Geometric Theory of the Recursion Operators for the Generalized Zakharov-Shabat System in Pole Gauge on the Algebra sl(n,C)

We consider the recursion operator approach to the soliton equations related to the generalized Zakharov-Shabat system on the algebra sl(n,C) in pole gauge both in the general position and in the presence of reductions. We present the recursion operators and discuss their geometric meaning as conjugate to Nijenhuis tensors for a Poisson-Nijenhuis structure defined on the manifold of potentials.


Introduction
The theory of nonlinear evolution equations (NLEEs) of soliton type (soliton equations or completely integrable equations) has developed considerably in recent decades. Interest in it is still big and there is a wide variety of approaches to these equations. However, some properties are fundamental to all approaches and one is that these equations admit the so called Lax representation, namely [L, A] = 0. In the last expression L and A are linear operators on ∂ x , ∂ t depending also on some functions q α (x, t), 1 ≤ α ≤ s (called 'potentials') and a spectral parameter λ. Since the Lax equation [L, A] = 0 must be satisfied identically in λ, it is equivalent to a system (in the case when A depends linearly on ∂ t ) of the type In most of the approaches the linear problem Lψ = 0 (auxiliary linear problem) remains fixed and the evolution equations (of a certain form) that can be obtained by changing the operator A are considered. The hierarchies of equations we obtain by fixing L are called the nonlinear evolution equations (NLEEs), or soliton equations, associated with (or related to) L (or with the linear system Lψ = 0). The hierarchies usually are named for some of the remarkable equations contained in them. The schemes according to which one can calculate the solutions to the soliton equations may be very different, but the essential fact is that the Lax representation permits one to pass from the original evolution defined by the system of equations (1) to the evolution of some spectral data related to the problem Lψ = 0. Since finding the spectral data evolution usually is not a problem, the principal difficulty is to recover the potentials from the spectral data. This process is called the inverse scattering method, which is described in detail in the monographs [4,10].
The generalized Zakharov-Shabat system (GZS system) we see below is one of the best known auxiliary linear problems. It can be written as follows Lψ = (i∂ x + q(x) − λJ) ψ = 0. (2) Here q(x) and J belong to a fixed simple Lie algebra g in some finite-dimensional irreducible representation. The element J is regular, that is the kernel of ad J (ad J (X) ≡ [J, X], X ∈ g) is a Cartan subalgebra h ⊂ g. The potential q(x) belongs to the orthogonal complement h ⊥ of h with respect to the Killing form and therefore q(x) = α∈∆ q α E α where E α are the root vectors; ∆ is the root system of g. The scalar functions q α (x) (the 'potentials') are defined on R, are complex valued, smooth and tend to zero as x → ±∞. We can assume that they are Schwartz-type functions. The classical Zakharov-Shabat system is obtained for g = sl(2, C), J = diag(1, −1).

Remark 1.
We assume that the basic properties of the semisimple Lie algebras (real and complex) are known and we do not give definitions of all the concepts related to them. All our definitions and normalizations coincide with those in [15].
Remark 2. When generalized Zakharov-Shabat systems on different algebras are involved we say that we have a generalized Zakharov-Shabat g-system (or generalized Zakharov-Shabat on g) to underline the fact that it is on the algebra g. When we work on a fixed algebra its symbol is usually omitted.
Here we may mention also that, in case when the element J is complex, the problem (2) is referred as a Caudrey-Beals-Coifman system [2] and only in the case when J is real it is called a generalized Zakharov-Shabat system. The reason for the name change is that the spectral theory of L is of primary importance for the development of the inverse scattering techniques for L and the cases when J is real or complex are quite different from the spectral viewpoint, see for example [5,11] in which the completeness of the so-called adjoint solutions of L when L is considered in an arbitrary faithful representation of the algebra g is proved. Referring for the details to the above work we simply remind the reader that the adjoint solutions of L are functions of the type w = mXm −1 where X is a constant element from g and m is a fundamental solution of Lm = 0. Let us denote the orthogonal projector (with respect to the Killing form (3)) on h ⊥ by π 0 . Then, of course, the orthogonal projector on h will be equal to id −π 0 . Further, let us put w a = π 0 w and w d = (id −π 0 )w. One of the most important facts from the theory of GZS systems is that if a suitable set of adjoint solutions (w i (x, λ)) is taken, then, roughly speaking, for λ on the spectrum of L the functions (w a i (x, λ)) form a complete set in the space of potentials. If one expands the potential over the subset of the adjoint solutions the coefficients are one of the possible minimal scattering data sets for L. Thus, passing from the potentials to the scattering data can be considered as a sort of Fourier transform, called a generalized Fourier transform. For it w a i (x, λ) play the role the exponents play in the usual Fourier transform. They are called generalized exponents or by abuse of language also adjoint solutions. Those familiar with the theory in sl(2; C) know that originally they were called 'squares' of the solutions of Lψ = 0. This interpretation of the inverse scattering transform was given for the first time in [1] and after that has been developed in a number of works, see, for example, the monographs [10,17] for complete study of sl(2, C)-case and for comprehensive bibliographies, and [2,11] for more general situations.
However, since in this article we shall not deal with the spectral properties of L, we shall call it a generalized Zakharov-Shabat system in all cases.
The recursion operators (generating operators, Λ-operators) are the operators for which the functions w a i (x, λ) are eigenfunctions and therefore for the generalized Fourier transform they play the same role as the differentiation operator in the usual Fourier transform method. For that reason recursion operators play central role in the theory of soliton equations -it is a theoretical tool which, apart from explicit solutions, can give most of the information about the NLEEs [10,36]. The theory of these operators is an interesting and developing area. Through them one may obtain: i) the hierarchies of the nonlinear evolution equations solvable through L; ii) the conservation laws for these NLEEs; iii) the hierarchies of Hamiltonian structures for these NLEEs.
It is not hard to find that the recursion operators related to L have the form, see [11] or the book [10] and the numerous references therein, Here, of course, ad q (X) = [q, X] and X is a smooth, rapidly decaying function with values in h ⊥ . The name 'recursion operators' has the following origin. Suppose we are looking for the NLEEs that have Lax representation [L, A] = 0 with L given in (2) and A of the form Then from the condition [L, A] = 0 we first obtain A n−1 = ad −1 J [q, A n ] and for 0 < k < n − 1 the recursion relations where Λ ± are as in (4). This leads to the fact that the NLEEs related to L can be written in one of the following equivalent forms: Remark 3. Strictly speaking, this is not the most general form of the equations solvable through L. Considering the right-hand side of the equations of the type ad −1 J q t = F n (q) as vector fields, in order to obtain the general form of the NLEEs associated with L one must take an arbitrary finite linear combination F of the vector fields F n with constant coefficients and write ad −1 J q t = F (q). We refer to (5) as the general form of the equations solvable through L for the sake of brevity.
There is another important trend in the theory of the GZS system and consequently for the recursion operators related to it. It turns out that this system is closely related to another one, called the GZS system in pole gauge (then the system L we introduced is called GZS system in canonical gauge). In order to introduce it, denote the group that corresponds to the algebra g by G. Then the system we are talking about is the following (with appropriate conditions on S(x) when x → ±∞), where O J is the orbit of the adjoint action of the group G, passing through the element J ∈ g. A gauge transformation of the type ψ → ψ −1 0 ψ =ψ where ψ 0 is a fundamental solution to the GZS system corresponding to λ = 0 takes the system L into the systemL if we denote S = ψ −1 0 Jψ o . One can choose different fundamental solutions ψ 0 and one will obtain different limiting values for S when x → ±∞ but usually for ψ 0 is taken the Jost soliton that satisfies lim x→−∞ ψ 0 = 1. The GZS system in pole gauge is used as the auxiliary linear problem to solve the equations that are classical analogues of equations describing waves in magnetic chains. For example, in the case of g = sl(2, C) (spin 1/2), one of the NLEEs related toL is the Heisenberg ferromagnet equation In the case of sl(3, C) the linear problemL is related to a classical analog of equations describing the dynamics of spin 1 particle chains [3,31]. The theory of NLEEs related to the GZS auxiliary system in canonical gauge (L) is in direct connection with the theory of the NLEEs related with the GZS auxiliary system in pole gauge (L). The NLEEs for both systems are in one-to-one correspondence and are called gaugeequivalent equations. This beautiful construction was used for the first time in the famous work of Zakharov and Takhtadjan [37], in which the gauge-equivalence of two famous equations -the Heisenberg ferromagnet equation (7) and the nonlinear Schrödinger equation was proved.
In fact the constructions for the system L and its gauge equivalentL are in complete analogy. Instead of the fixed Cartan subalgebra h = ker ad J , we have a 'moving' Cartan subalgebra h S = ker ad S(x) ; a 'moving' orthogonal (with respect to the Killing form) complementary space h ⊥ S to h S etc. We have the corresponding adjoint solutionsm =ψXψ −1 , whereψ is a fundamental solution ofLψ = 0 and X is a constant element in g. If we denote bym a andm d the projections ofm on h ⊥ S and h S respectively, then the corresponding recursion operators are constructed using the fact that the functionsm a must be eigenfunctions for them. The evolution equations associated with the system (6) and gauge-equivalent to the equations (5) have the form whereΛ ± are the recursion operators forL and π S is the orthogonal projector on h ⊥ S . One can see that where Ad is denotes the adjoint action of the simply connected Lie group G having g as algebra.
Remark 4. In order to understand why in the hierarchy (8) appears π S A n one must mention that one can prove that for any constant H ∈ h So for the GZS system in pole gauge everything could be reformulated and the only difficulty is to calculate all the quantities that are expressed through q and its derivatives through S and its derivatives. Though in each particular case the details may be different there is a clear procedure to achieving that goal. The procedure was developed in the PhD thesis [30], outlined in [12,14] for the sl(2, C) case, and for more general cases in [13]. In the case sl(3, C) the procedure has been carried out in detail in [31]. The theory of the recursion operator for the GZS system in canonical gauge in the presence of the Mikhailov type reductions has been also considered, see for example [6] for a treatment from the spectral theory viewpoint, or [33] for geometric treatment.
The theory for the system in pole gauge in presence of Mikhailov type reductions has been also the subject of recent research. In [7,8] the case sl (3, C) in the presence of Z 2 × Z 2 reduction was considered (Gerdjikov-Mikhailov-Valchev system or GMV system). In [34] it was shown that the operators found in [8] using classical technique and the technique developed in [16], are restrictions of the general position recursion operators for sl(3, C), on certain subspaces of functions. In [35] the geometric theory behind the recursion operators for the GMV system was presented. In the present article we shall generalize the theory developed in [33,34,35] for the algebra sl(3, C) to the algebra sl(n, C). Another new feature is that we discuss here in more depth the fundamental fields of the Poisson-Nijenhuis structure (P-N structure) related to the GMV system and its generalizations.
2 Recursion operators for the GZS sl(n, C)-system in pole gauge

Preliminary results
Let us consider the GZS system in canonical gauge in the case g = sl(n, C) -the algebra of all traceless n × n complex matrices. The Cartan subalgebra h consists of all traceless diagonal matrices and the space h ⊥ of all off-diagonal matrices. The Killing form can be expressed through the trace form, we have X, Y = 2n tr XY for any X, Y ∈ sl(n, C), see [15]. Since the element J belongs to h it has the form J = diag(λ 1 , λ 2 , . . . , λ n ), where n k=1 λ k = 0. Next, J is regular, which means that for any i = j we have λ i = λ j . In that case J 'generates' the Cartan subalgebra h in the following sense. Consider the matrices Since tr J k = 0 for k = 1, 2, . . . , n these matrices belong to h. One can easily show that these matrices are linearly independent and hence generate the Cartan subalgebra. The same can also be deduced if one calculates the determinant of the Gram matrix Another observation that we want to make is that J and S satisfy the equation (Since λ i are the eigenfunctions of J and S the above is just the Cayley-Hamilton theorem.) This equation can be written in the form where the coefficients a s are homogeneous polynomials of degree s in λ 1 , λ 2 , . . . , λ n . If necessary, one can express them in terms the symmetric polynomials J s , J s , i.e., polynomials in λ i , 1 ≤ i ≤ n, using the Newton formulae but it is not needed for our purposes. We only note that a 1 = tr J = 0. At the end of our preparations let us denote S s;x = ∂ x S s . Then we have: Proof . We need to prove that for all 1 ≤ k, l ≤ n − 1 we have S k , S l;x = 0. Since S k = S k − (tr J k /n)1 we must prove that tr(S k (S l ) x ) = 0. Using the properties of the trace we see that this is equivalent to tr(S k+l−1 S x ) = 0. On the other hand, for any integer m > 0 we have tr S m = tr J m = const. Therefore, ∂ x tr S m = m tr S m−1 S x = 0 and our result follows.

Calculation of the recursion operators
Now we pass to the calculation of the recursion operator(s)Λ ± for the GZS system in pole gauge on sl(n, C). We shall use the equation which is satisfied by every function of the typew =ψA(ψ) −1 , where A is a constant matrix andψ is a fundamental solution of (6). We havẽ (Note that ker ad S = h S , the space h ⊥ S is its orthogonal space with respect to the Killing form and by upper indices 'h' and 'a' we denote projections onto these spaces.) We have seen that the matrices or in other words, In order to find the coefficients a k we calculate the inner product of the left hand side of (10) with S j . Then, taking into account Proposition 1, we arrive at the following system Assuming that for the eigenfunctions ofΛ + we have lim x→+∞ a s = 0 and for the eigenfunctions where ∂ −1 x stands for one of the two operators Consequently, inserting the functions a k into (10), we obtainΛ ± (w a ) = λw a wherẽ or equivalently, The above operators act on functionsZ(x) that are smooth, rapidly decaying and such that (11), (12) give us the recursion operators but they can be written in more concise form if we introduce: • The row vectors • The column vectors Then the recursion operators acquire the form It is easy to check now that the recursion operators for the GZS systems on the algebras sl(2, C), sl(3, C) (see [12,31,32]) are obtained from the general expressions (13), (14) as particular cases.
The last thing that remains to be done is to express the operator ad −1 S through S. For this note that, if all the eigenvalues of J are different, the operators ad J and ad −1 J considered on h ⊥ (ad S and ad −1 S considered on h ⊥ S respectively) are simple and have common eigenvectors. Then we can apply the following proposition which is actually the spectral decomposition theorem for a given simple matrix A, see [18].
Then the matrix f (A) has as eigenvalues µ 1 , µ 2 , . . . , µ m and the same eigenvectors as A and the polynomial f (λ) is the polynomial of minimal degree having that property.
Remark 5. It is not difficult to see that l k (A) is the projector onto the subspace corresponding to the eigenvalue λ k in the splitting of the space into eigenspaces of the matrix A.
Remark 6. In case the matrix A is not simple one again can produce a polynomial f (λ) of minimal degree having the property stated in the Proposition 2 though its construction is more complicated, see [18]. In case just ad −1 J is needed, one can also use the following procedure: 1. The minimal polynomial m(λ) of ad J (on the whole algebra) is a product m(λ) = λm 1 (λ) and λ and m 1 (λ) are co-prime.
2. The algebra splits into direct sum of invariant subspaces ker ad J and im ad J (because ad J is skew-symmetric) and the minimal polynomials of ad J on these spaces are λ and m 1 (λ) respectively.
3. On im ad J the operator ad J is invertible and one can find ad −1 J as polynomial in ad J multiplying the equation m 1 (ad J ) = 0 by ad −1 J .
Note that the polynomial g(λ) such that ad −1 J = g(ad J ) will have now degree deg(m 1 ) − 1.
In the case of an arbitrary semisimple Lie algebra, assuming that all the values α(J), α ∈ ∆ are different, the operators ad J and ad −1 J have eigenvalues α(J) and 1/α(J), α ∈ ∆ respectively and eigenvectors E α , α ∈ ∆. Then ad −1 J is equal to l(ad J ), where l(λ) is the polynomial In our case the roots are and the expression for ad −1 S can be written easily, we shall do it a little further. What we obtained already allows us to make an important observation. As the polynomial l(λ) from (15) is of the form λl 0 (λ), where l 0 is another polynomial, then we have l(ad J ) = l 0 (ad J ) ad J . Therefore we have l(ad J )π 0 = l(ad J ).
In the same way Therefore, if one assumes that ad −1 S is given by a polynomial in ad S , writing the projector π S in the expression for the recursion operators (13), (14) is redundant.
Continuing our discussion about ad −1 S , as we already remarked, in the case when we need only to express ad −1 S through ad S , instead of Proposition 2 we can use the minimal polynomial for ad S (restricted to h ⊥ of course). Indeed, in the case of an arbitrary semisimple Lie algebra g and regular J let us assume that all the values α(J) are different. Then the minimal polynomial for ad J on h ⊥ has the form Since m(ad S ) = 0 we obtain where R(λ) is the polynomial Both expressions (16) and (17) for ad −1 S give the same result in the case where all α(J) are different. Indeed, both polynomials λl(λ) and λR(λ) are monic and when λ = α(J), α ∈ ∆ give 1/α(J). Since they are of degree 2p − 1 these polynomials coincide. In particular, in the case of the algebra sl(n, C) we get Unfortunately, the expressions for ad −1 S become very complicated for big n, hampering the possible applications. A simplification can be obtained for some particular choices of J, for which the minimal polynomial of ad J on h ⊥ has smaller degree than in the case of general J.
Here are the expressions for ad −1 S used up to now in the literature: The last condition ensures that all α(J)'s are different [31,32] where 3. GZS type system on g = sl(3, C) with Z 2 ×Z 2 reduction (GMV system), J = diag(−1, 0, 1) [8,32] ad −1 S = l(ad S ), In this case the fact that some of the eigenvalues of ad J on h ⊥ (and consequently of ad S on h ⊥ S ) are not simple leads to a decrease in the degree of the polynomial l(λ).

Geometric interpretation
Fixing the element J for the GZS g-system in pole gauge, the smooth function S(x) with domain R, see (6), is not subject to any restrictions except that S(x) ∈ O J and S(x) tends fast enough to some constant values when x → ±∞. First let us consider the even more general case when S(x) is smooth, takes values in g and when x → ±∞ tends fast enough to constant values. The functions of this type form an infinite-dimensional manifold which we shall denote by M.
Then it is reasonable to assume that the tangent space T S (M) at S consists of all the smooth functions X : R → g that tend to zero fast enough when x → ±∞. We denote that space by F(g). We shall also assume that the 'dual space' T * S (M) is equal to F(g) and if α ∈ T * S (M), where , is the Killing form of g.

Remark 7.
In other words, we identify T * S (M) and T S (M) using the bi-linear form , . We do not want to make the definitions more precise, since we speak rather about geometric picture than about precise results. Such results could be obtained after a profound study of the spectral theory of L andL. In particular, we have put the dual space in quotation marks because it is clearly not equal to the space of continuous linear functionals on F(g). We also emphasize that when we speak about 'allowed' functionals H on M we mean that δH δS ∈ T * S (M) ∼ F(g).
Now we want to introduce some facts. The first fact is that since we identify T * S (M) and T S (M) the operators can be interpreted as Poisson tensors on the manifold M. This is well known, see for example [10], where the issue has been discussed in detail and the relevant references are given. One can also verify directly that if H 1 , H 2 are two functions (allowed functionals) on the manifold of potentials M then are Poisson brackets. It is also known from the general theory that these Poisson tensors are compatible [10,Chapter 15]. In other words P + Q is also a Poisson tensor. Note that the tensor Q is the canonical Kirillov tensor which acquires this form because the algebra is simple and consequently the coadjoint and adjoint representations are equivalent. Now let O J be the orbit of the adjoint action of G passing through J. Let us consider the set of smooth functions f : R → O J such that when x → ±∞ they tend fast enough to constant values. The set of these functions is denoted by N and clearly can be considered as a submanifold of M. If S ∈ N the tangent space T S (N ) consists of all smooth functions X that vanish fast enough when x → ±∞ and are such that X(x) ∈ T S(x) (O J ). (Recall that O J is a smooth manifold in the classical sense). We again assume that T * S (N ) ∼ T S (N ) and that these spaces are identified via , .
We can try now to restrict the Poisson tensors P and Q from the manifold M to the manifold N . The problem how to restrict a Poisson tensor on a submanifold has been solved in principle [24], see also [27,28]. We shall use a simplified version of these results (see [22,23]) and we shall call it the first restriction theorem: Then there exists unique Poisson tensorP onM, j-related with P , that is The proof of the theorem is constructive. One takes β ∈ T * m (M), then represents (j * β) m as α 1 + α 2 where α 1 ∈ X * P (M) m , α 2 ∈ T ⊥ (M) m and putsP m (β) = P m (α 1 ) (we identify m and j(m) here).
The restriction we present below has been carried out in various works in the simplest case g = sl(2, C), see for example [23]. We do it now in the case g = sl(n, C). Restricting the Poisson tensor Q is easy, one readily gets that the restrictionQ is given by the same formula as before The tensor P is a little harder to restrict. Let us introduce some notation and facts first. Since J is a regular element from the Cartan subalgebra h, each element S from the orbit O J is regular. Therefore h S ≡ ker ad S is a Cartan subalgebra of sl(n, C) and we have If X ∈ T S (N ) then X(x) ∈ h ⊥ S (x) and X vanishes rapidly when x → ±∞. We shall denote the set of these functions by F(h ⊥ S ) so X ∈ F(h ⊥ S ) (this means a little more than simply X ∈ h ⊥ S ). Using the same logic, for X ∈ F(h ⊥ S ) we write ad S (X) ∈ F(h ⊥ S ) which means that the function ad S(x) X(x) belongs to F(h ⊥ S ). Now we are in a position to perform the restriction of P on N . For S ∈ N we have We see that T ⊥ (N ) S is the set of smooth functions α such that α ∈ h S and such that they vanish fast enough when x → ±∞. We shall denote this space by F(h S ). We introduce also the space of functions X ∈ h S which tend rapidly to some constant values when x → ±∞ and denote this space by F(h S ) 0 . Clearly, since S k , k = 1, 2, . . . , n − 1 span h S , we have In what follows we shall adopt matrix notation and shall denote by A(x) the column with components a k (x). Then X * P (N ) S ∩ T ⊥ (N ) S consists of the elements with a k (x) vanishing at infinity and such that i∂ so we must have A x = 0 and a s (x) are constants. But a s (x) must also vanish at infinity so we see that a s = 0. Consequently X * P (N ) S ∩ T ⊥ (N ) S = {0} ⊂ ker P S . Let us take now arbitrary α ∈ T * (N ) S . We want to represent it as α 1 + α 2 , α 1 ∈ X * (N ) S , α 2 ∈ T ⊥ (N ) S . First of all, α 2 = SB(x) where B(x) a column with components b s (x) -scalar functions that vanish at infinity. This means that where i∂ x α 1 ∈ F(h ⊥ S ). But then we have where G is the Gram matrix we introduced earlier. Therefore Remark 8. In the theory of recursion operators when one calculates the hierarchies of NLEEs or the conservation laws the expressions on which the operator ∂ −1 x acts are total derivatives. Thus the same results will be obtained choosing for ∂ −1 x any of the following operators However, more frequently one uses the third expression when one writes the corresponding Poisson tensors in order to make them explicitly skew-symmetric.
Returning to our task, let us put where α 1 and α 2 lie in the spaces X * (N ) S and T ⊥ (N ) S = F(h S ) respectively. We note also that dj * S β = π S (β). Thus the conditions of the first restriction theorem are fulfilled and if β ∈ T * S (N ) the restrictionP of P on N has the form The tensorQ is invertible on N , so one can construct a Nijenhuis tensor N =P • ad −1 Taking into account that ad −1 S (X), S = 0, the above can be cast into the equivalent form From the general theory of compatible Poisson tensors it now follows that Theorem 2. The Poisson tensor fieldQ (20) and the Nijenhuis tensor field N given by (21), (22) endow the manifold N with a P-N structure.
The final step is to calculate the dual of the tensor N with respect to the pairing , . A quick calculation, taking into account that ad S is skew-symmetric with respect to the Killing form, gives or equivalently, But these are the recursion operatorsΛ ± for the GZS system in pole gauge from (13), (14) and our results confirm the general fact that the recursion operators and the Nijenhuis tensors are dual objects. Now, according to the general theory of recursion operators, see [10], the NLEEs related to the systemL have the form where H is an element of the Cartan subalgebra of sl(n, C). As discussed earlier, π S can be expressed as a polynomial in ad S . Then (23) gives the hierarchies of NLEEs related to the GZS system in pole gauge as hierarchies of equations gauge-equivalent to hierarchies related to the GZS system in canonical gauge. If, however, one is not interested in finding the pairs of gauge-equivalent equations, but wants just to find the NLEEs, one can proceed as follows.
Recall that recursion operators also produce the hierarchy of Lax pairs. In fact, if a NLEE has Lax representation [L,Ã] = 0 withL in general position andÃ has the form Using a gauge transformation depending only on t one can ensure that A 0 = 0 and then the coefficientsÃ k for k = 1, 2, . . . , n − 1 may be calculated with the help of the recursion operator in the following waỹ For the coefficientÃ n−1 one has that i∂ xÃn = [S,Ã n−1 ] and, sinceÃ n ∈ h S , there are n − 1 scalar functions a k (x) forming a column vector a(x) such that A n = S(x)a(x). This gives and therefore a x = 0 so a k (x) are constants. ThusÃ n−1 = i ad −1 S (Sa) and the hierarchy of the NLEEs related toL is = (a 1 , a 2 , . . . , a n−1 ) t .

Algebraic aspects
As a matter of fact the situation most interesting for applications is when we do not have the GZS system in pole gauge in the general case, but when additional restrictions are made. For example, in the Heisenberg ferromagnet equation we require that S = S † (where † stands for Hermitian conjugation). Similarly, for systems describing spin chain dynamics that we mentioned in the case of sl(3, C), we also require S † = S. All this means that we require for n = 2, 3 that S belongs to i su(n). The algebra su(n) is a real form of sl(n, C) with respect to complex conjugation σ(X) = −X † and sl(n, C) = su(n) ⊕ i su(n), [su(n), su(n)] ⊂ su(n), The space i su(n) is the space of n × n Hermitian matrices. Of course, it is not a real Lie algebra since, if X, Y ∈ i su(n), then [X, Y ] ∈ su(n) and hence i[X, Y ] ∈ i su(n). Next, since the Cartan subalgebra h is invariant under σ, it also splits h = (h ∩ su(n)) ⊕ (h ∩ i su(n)).
The first space consists of diagonal matrices with purely imaginary entries, while the second consists of diagonal matrices with real entries. If we want S to be Hermitian, then it is natural to assume that J is real. In fact the reduction one obtains in this way is most effectively treated by the notion of Mikhailov's reduction group, see [19,25,26]. According to that concept, in order to perform a reduction we must have a group G 0 acting on the fundamental solutions of the linear problems L and A in the Lax representation. Assume we have the GZS system in pole gauge and we take the group generated by one element g 1 , acting on the fundamental solutions as where * stands for complex conjugation. Since g 2 1 = id, the group generated by g 1 is isomorphic to Z 2 and we have Z 2 -reduction. The invariance of the set of fundamental solutions under the action (24) means that σ(S) = −S (that is S ∈ i su(n)) and that S belongs to the orbit of SU(n) passing through J ∈ i su(n). Let us denote this orbit by O J (SU(n)). Thus J must be real. Moreover, if the operator A has the form then, if the set of fundamental common solutionsψ of (6) and Aψ = 0 is G 0 -invariant, one must have also σ(A k ) = A k where σ is the complex conjugation introduced earlier, that is A k ∈ i su(n). One may note that the generating operators are the same as before simply we may assume that all matrices are in i su(n).
One can have also another complex conjugation of sl(n, C). Denote it by τ . Then one can construct another Z 2 reduction. One can, for example, take τ (X) = X * and g acting as g(ψ(x; λ)) = (ψ(x; λ * ) * .
Then one obtains that S * = −S, A * k = −A * k , so introducing S 0 = iS and A 0 k = iA k one sees that (after canceling i) the L, A pair is real, with matrices belonging to sl(n, R).
It is possible to have restrictions defined by two complex conjugations. This can be achieved in the following way. Suppose τ is another complex conjugation, commuting with σ. Then as it is well known h = στ = τ σ is an involutive automorphism of the algebra sl(n, C). In fact doing the things the other way round is easier -to find an involutive automorphism h commuting with σ and then put τ = hσ = σh. One can take, for example, h(X) = J K XJ K , where J K is a diagonal matrix diag(1, 1, . . . , 1 K times , −1, −1, . . . , −1 n−K times ), and one can put τ = hσ = σh. Then one can consider the group G 0 with generating elements g 1 (as in (24)) and g 2 , where Since g 1 g 2 = g 2 g 1 , g 2 1 = g 2 2 = id the reduction group G 0 is isomorphic to Z 2 × Z 2 so the invariance of the fundamental solutions with respect to the actions (24) and (25) defines Z 2 × Z 2 reduction. In [8,9] the above reduction has been applied in the case of the algebra sl(3, C) with J 1 = diag(1, −1, −1) (here K = 1). A similar reduction group but this time related to the algebra so (5) has been considered recently in [29].
Returning to the general case, note that instead of the generators g 1 , g 2 we could use g 1 , g 1 g 2 and The invariance with respect to the group generated by g 1 , g 2 means that or equivalently The complex conjugation τ also splits the algebra sl(n, C) (On s the map τ is equal to id, on is to − id.) We prefer to work with the involutive automorphism h instead of τ . Then the algebra splits into two invariant subspaces for h and these spaces are orthogonal with respect to the Killing form. We have A calculation shows that the subalgebra f 0 consists of matrices U 0 having block form where tr A + tr B = 0, and the diagonal blocks A and B have dimensions K × K and (n − K) × (n − K) respectively. In terms of the same type of block matrices the space f 1 consists of matrices of the type Since σ and h commute the spaces of the real and purely imaginary elements for σ -namely su(n) and i su(n), are split by h into two subspaces su(n) = (f 0 ∩ su(n)) ⊕ (f 1 ∩ su(n)), i su(n) = (f 0 ) ⊕ (f 1 ).
As we shall see S(x) ∈ f 1 so the space f 1 is of particular interest to us. As is easily seen, this space consists of matrices of the form while f 0 ∩ i su(n) consists of matrices of the form Therefore the real vector space f 1 ∩ i su(n) can be considered as isomorphic to the quotient space su(n)/(s(u(K) × u(n − K))).
Below, since all the matrices S k , S k;x belong to i su(n) (that is, they are Hermitian), we shall not write explicitly that they belong to i su(n) and shall concern ourselves only with whether they belong to f 0 , f 1 , h S and h ⊥ S . This makes the formulas simpler and is possible due to the facts that restricting to i su(n) does not change the form of the recursion operators and the restriction of the Killing form to i su(n) is again nondegenerate. Now, since h(S) = −S we have ad S •h = −h • ad S and the Cartan subalgebra h S = ker ad S also splits into two subspaces orthogonal to each other Since orthogonality with respect to the Killing form is preserved by h, the space h ⊥ S is also invariant under h and Consider now S = S 1 , S 2 , . . . , S n−1 , the basis of h S we introduced earlier. From the above it follows that and one can write Consequently, In addition, since f 0 and f 1 are orthogonal with respect to the Killing form, we have that S 2s , S 2k−1 = 0.
Finally, one immediately sees that Let us introduce the following spaces: 1. F( 0 h ⊥ S ) consists of all smooth, rapidly decaying functions X(x) on the line such that consists of all smooth, rapidly decaying functions X(x) on the line such that Naturally, where the spaces are orthogonal with respect to the form · .
Remark 9. Note that all our matrices are also elements from i su(n) so strictly speaking S k are elements from h 0 S ∩ i su(n) or h 1 S ∩ i su(n) and S k;x are elements from 0 h ⊥ S ∩ i su(n) or 1 h ⊥ S ∩ i su(n) but we agreed not to write i su(n) as this will make the notation even more complicated.
After these preliminaries, assuming that all the quantities are as above we have Proposition 3. The recursion operatorsΛ ± interchange the spaces F( 0 h S ) and F( 1 h S ) in the sense that Proof . In order to make the calculation easier, let us introduce the rows of elements: where 2k − 1 is the largest odd number less then or equal to n and Since J m , J n = S m , S n these matrices have constant entries. With the help of the matrices (27), (28) the recursion operators can be written into the form Assume thatZ ∈ F( 0 h ⊥ S ). Then since Z , S 0 = Z , S 1 = 0 and Z , S 1;x = 0 we get that Recalling (26) we get thatΛ ± (Z) ∈ F( 1 h ⊥ S ). The above also means that trough the expressions (29), (30) and we get thatΛ ± (Z) ∈ F( 0 h ⊥ S ). In the same way as above (31), (32) define operatorsΛ 1 ±the restrictions ofΛ ± on F( 1 h ⊥ S ). In other words we havẽ In similar situations (when we have Z 2 reductions) the operatorsΛ 0 ± ,Λ 1 ± are considered as factorizing the recursion operator. In some sense this is true, since if one considers (Λ ± ) 2 acting on F( 0 h ⊥ S ) we can write it asΛ 1 ±Λ 0 ± . On the other hand (Λ ± ) 2 acting on F( 1 h ⊥ S ) can be written asΛ 0 We think that is more accurate to treat the operatorsΛ 0 ± ,Λ 1 ± as restrictions of the operatorsΛ ± . The geometric picture we are going to produce below also supports this viewpoint.

Geometric aspects
The geometric situation in the presence of reductions is also interesting. The point is that the canonical Poisson structureQ S = ad S simply trivializes, apparently destroying the geometric interpretation given in the case of the GZS pole gauge system in general position. Indeed, under the restrictions considered in this section we first note that the space on which the 'point' S(x) takes its values is iO J (SU(n)) where J is real (and regular). As remarked already this simply makes all the matrices Hermitian and everything remains as it was. However, imposing the second Z 2 reduction (the one defined by the automorphism h) means that S(x) belongs to the space of matrices taking values in f 1 and such that they converge rapidly to some constant values when x → ±∞. So for the manifold of potentials Q we have: Then the tangent space T S (Q) to the manifold (33) at the point S is the space F( 1 h ⊥ S ) (for the sake of brevity we again 'forget' to mention that the tangent vectors must be also elements of i su(n)). But because ad S interchanges F( 1 h ⊥ S ) and F( 0 h ⊥ S ) and they are orthogonal with respect to , forZ 1 ,Z 2 ∈ T S (Q) we have that Z 1 , [S,Z 2 ] = 0, so indeed the tensorQ becomes trivial. Let us see what happens with the Nijenhuis tensor. Writing everything with the notation we introduced in this section we have Again we can assume that the right-hand sides of the above equations define the operators N 1 , N 2 and An immediate calculation shows that for the conjugates of the operators N 0 , N 1 with respect to , we have Remark 10. One should bear in mind that ∂ −1 x that enters in the expressions for N 0 , N 1 is treated either as x −∞ · dy or as x +∞ · dy.
One can easily see that N (T S (Q)) does not belong to T S (Q), so the restriction of N on Q cannot be a Nijenhuis tensor on Q. However, so N 2 becomes a natural candidate. Indeed, let us recall the following facts from the theory of P-N manifolds, see [10,20,21,22]: If M is a P-N manifold endowed with Poisson structure P and Nijenhuis tensor N , then for k = 1, 2, . . . each pair (N k P = P (N * ) k , N s ) also endows M with a P-N structure. ii) The tangent spaces ofM, considered as subspaces of the tangent spaces of M are invariant under N , so that N allows a natural restrictionN toM, that isN is j-related with N .
Then (P ,N ) endowM with a P-N structure.
We call the above theorem the second restriction theorem.
In view of what we have already, we need only find the restrictionP ofP on Q and thenP and the restriction of N 2 will endow Q with a P-N structure. Thus we have the following candidates for restriction -the Poisson tensorP = N • ad S = ad S •N * and the Nijenhuis tensor N 2 (or N −2 ). Let us takeP and try to restrict it.
We want to apply the first restriction theorem. X * (P ) S consists of smooth functions β, going rapidly to zero when |x| → ∞ such that β(x) ∈ h ⊥ S (x) andP (β) ∈ T S (Q). The last means that that is, for arbitrary smooth function X(x) such that X ∈ f 0 S and going rapidly to zero when |x| → ∞ we have The space T ⊥ (Q) S consists of smooth β(x) such that β ∈ h ⊥ S , going rapidly to zero when |x| → ∞ and satisfying for each smooth function Y (x), Y ∈ f 1 S (x) going rapidly to zero when |x| → ∞. Arguments similar those used to prove Haar's lemma in the variational calculus show that for each x where Y (x) is as above. Then, if β ∈ X * (P ) S ∩ T ⊥ (Q) S , we shall have simultaneously (34) and (35) so β ∈ kerP S . Thus the first condition of the first restriction theorem is fulfilled. In order to see that the second condition also holds, we introduce Lemma 1. The operatorP has the properties The proof of the lemma is obtained easily since the spaces f 0 S and f 1 S are invariant with respect to π S .
Using the lemma, suppose β ∈ h ⊥ S is a smooth function going rapidly to zero when |x| → ∞. Clearly, we can write it uniquely into the form ThenP S (β 0 ) ∈ F(f 0 S ),P S (β 1 ) ∈ F(f 1 S ) and we see thatP S β 0 ∈ X * (P ) S ,P S β 1 ∈ T ⊥ (Q) S . So the second requirement of the first restriction theorem is also satisfied andP allows restriction. If γ is a 1-form on Q, that is γ ∈ F(f 1 S ), the restrictionP is given bȳ manifolds, if X is a fundamental field then all the fields N p X, p = 1, 2, . . . are also fundamental and commute, that is they have zero Lie brackets. Thus to each H ∈ h corresponds a hierarchy of fundamental fields N k [H, S] and we obtain n−1 independent families of fundamental fields. For different p and different H ∈ h the fields from these families also commute. As a matter of fact this is commonly referred to as 'the geometry' behind the properties of the hierarchies (8). We can cast these hierarchies in a different form. Using the expression (21) and taking into account that ad −1 S ad S H = π S (H) after some simple transformations in which we use the properties of the Killing form and Proposition 1 we get N (X H )(S) = iπ S ∂ x (π S (H)) − iS x G −1 ∂ −1 x ∂ x (π S (X)), S(x) = −iS x G −1 H, S(±∞) = −S x G −1 H, iS(±∞) .
The last expression shows that N (X H )(S) is a linear combination of the fields S 1;x = S x , S 2;x = (S 2 ) x , . . . , S n−1;x = (S n−1 ) x . Since the evolution equations corresponding to the fields X H are linear one can say that the hierarchies of the fundamental fields that correspond to nonlinear equations are generated by the above vector fields. Now, let us assume that one has reductions as in the above and let us first consider what happens when we restrict to i su(n). In that case we must take H ∈ su(n), that is, iH must be a diagonal, traceless real matrix. Now we can see that the coefficients in front of the fields S k;x are real since su(n) is a compact real form of sl(n) and on it the Killing form is real so both G and H, iS(±∞) are real. When we have the additional restriction defined by the automorphism h the situation is more complicated. Since h(S) = −S and h(H) = H for any H ∈ h we have h(X H ) = −X H so X H ∈ F( 1 h ⊥ S ). Thus N (X H ) ∈ F( 0 h ⊥ S ) and more generally for p = 0, 1, 2, . . .
Remark 11. The fact that N (X H ) ∈ F( 0 h ⊥ S ) can also be seen easily from (37). Indeed, because H is an element of the Cartan subalgebra it is orthogonal to all elements of f 1 and in (37) we have a linear combination of the elements S m with odd index m, that is −S x G −1 H, iS(±∞) = −S 1;x 1 G −1 H, iS 1 (±∞) .
Thus when we have a reduction defined by the automorphism h the first of the series in (38) consists of vector fields tangent to the manifold Q while the vector fields from the second series are not tangent to Q.
Naturally, the vector fields from the first series are the candidates for being fundamental fields of the P-N structure on Q and, discarding the first fields in the hierarchies, we see that we have fields of the type N 2p S 2j−1;x = (N 0 N 1 ) p S 2j−1;x .
These are not, however, all the fundamental fields that we can produce. The fields of the type N S 2j;x are fundamental for N and hence are fundamental for N 2 also. Besides, since S 2j;x ∈ F( 0 h ⊥ S ) we have N S 2j;x ∈ F( 1 h ⊥ S ). As easily checked, the fields N S 2j;x are tangent to Q. Thus, finally, the hierarchies of the the fundamental fields of the P-N structure when we restrict to Q are generated by the fields N 2p S 2j−1;x = (N 0 N 1 ) p S 2j−1;x , N 2p+1 S 2l;x = (N 0 N 1 ) p N 0 S 2l;x , 1 ≤ j ≤ k, 1 ≤ l ≤ s, p = 0, 1, 2, . . . , where k and s are such that 2k − 1 is the largest odd number less than n and 2s is the largest even number less than n. These numbers define the sizes of the Gram matrices 0 G and 1 G. For example, in the case of the algebra sl(3, C), we have the fields These fundamental fields give rise to hierarchies of integrable equations of the type S t = F (S) where F (S) is a finite linear combination of the fundamental fields (39), see [7].

Conclusions
We have been able to show that the geometric interpretation known for the the recursion operators related to the generalized Zakharov-Shabat system on the algebra sl(n, C) in pole gauge holds also in the presence of Z 2 ×Z 2 reductions of certain classes and we have explicitly calculated the recursion operators. It is an interesting question whether analogous results could be obtained for Zakharov-Shabat type systems on the other classical Lie algebras.