Symmetry, Integrability and Geometry: Methods and Applications The Non-Autonomous Chiral Model and the Ernst Equation of General Relativity in the Bidifferential Calculus Framework

The non-autonomous chiral model equation for an $m \times m$ matrix function on a two-dimensional space appears in particular in general relativity, where for $m=2$ a certain reduction of it determines stationary, axially symmetric solutions of Einstein's vacuum equations, and for $m=3$ solutions of the Einstein-Maxwell equations. Using a very simple and general result of the bidifferential calculus approach to integrable partial differential and difference equations, we generate a large class of exact solutions of this chiral model. The solutions are parametrized by a set of matrices, the size of which can be arbitrarily large. The matrices are subject to a Sylvester equation that has to be solved and generically admits a unique solution. By imposing the aforementioned reductions on the matrix data, we recover the Ernst potentials of multi-Kerr-NUT and multi-Demianski-Newman metrics.


Introduction
The bidifferential calculus framework allows to elaborate solution generating methods for a wide class of nonlinear "integrable" partial differential or difference equations (PDDEs) to a considerable extent on a universal level, i.e. resolved from specific examples. It takes advantage of the simple rules underlying the calculus of differential forms (on a manifold), but allows for a generalization of the latter, which is partly inspired by noncommutative geometry. For a brief account of the basic structures and some results we refer to [1] (also see the references therein), but all what is needed for the present work is provided in Section 2. In this framework we explore the non-autonomous chiral model equation for an m×m matrix g, where ρ > 0 and z are independent real variables, and a subscript indicates a corresponding partial derivative. It apparently first appeared, supplemented by certain reduction conditions (see Section 5), as the central part of the stationary axially symmetric Einstein vacuum (m = 2) and Einstein-Maxwell (m = 3) equations (see in particular [2,3,4,5,6,7,8]). For m > 3 this equation is met in higher-dimensional gravity, with a correspondingly enlarged number of Killing vector fields (see e.g. [9,10,11,12,13,14,15,16,17,18,19]). A version of the above equation also arises as the cylindrically symmetric case of the (2 + 1)-dimensional principal chiral model [20] and as a special case of the stationary Landau-Lifshitz equation for an isotropic two-dimensional ferromagnet [21]. The first construction of "multi-soliton" solutions of (1.1) has been carried out by Belinski and Zakharov [5,6] (also see [7]) using the "dressing method" 1 . Here (1.1) is expressed as the integrability condition of a linear system, which depends on a (spectral) parameter and involves derivatives with respect to the latter. Another approach is based on a linear system that depends on a variable spectral parameter, i.e. a parameter that depends on the variables ρ and z [3]. In Appendix B we show that both linear systems arise from a universal linear system (see Section 2) in the bidifferential calculus framework (also see [31] for a relation between the two linear systems).
In the present work, we concentrate on a surprisingly simple general solution generating result in the bidifferential calculus framework, which has already been successfully applied in various other cases of integrable (soliton) equations [1,32,33,34] to generate multi-soliton families. In order to make it applicable to the non-autonomous chiral model, a slight generalization is required, however (see Section 3 and Appendix A). Section 4 then elaborates it for the m × m non-autonomous chiral model. We obtain solutions parametrized by four matrices. Two of them arise as solutions of an n × n matrix version of the quadratic equation for pole trajectories that first appeared in the solution generating method of Belinski and Zakharov [5,6,7]. It then remains to solve a Sylvester equation, where two more matrices enter, which are constant of size m × n, respectively n × m. Since n can be arbitrarily large, we obtain an infinite family of solutions. The Sylvester equation is easily solved if the first two matrices are chosen diagonal, and in this case one recovers "multi-soliton" solutions. Additional solutions are obtained if the two n × n matrices are non-diagonal. In this case it is more difficult to solve the Sylvester equation, though a not very restrictive spectrum condition ensures the existence of a unique solution. Except for an example in Section 5, we will not elaborate this case further in this work.
Section 5 addresses reductions, in particular to the Ernst equation of general relativity. It turns out that the "multi-soliton" solutions of the stationary, axially symmetric Einstein vacuum and Einstein-Maxwell equations are indeed in the generated class of solutions of the nonautonomous chiral model. We thus obtain a new representation of these solutions. It has the property that the superposition of two (or more) "solitons" (e.g. black holes) simply corresponds to block-diagonal composition of the matrix data parametrizing the constituents. This puts a new perspective on an old result about one of the most important integrable equations in physics.
We would like to stress that the solutions of the non-autonomous chiral model and the Ernst equation(s), (re)derived in this work, originate from a universal result that also generates multi-soliton solutions of various other integrable equations in a non-iterative way. The crucial step is to find a "bidifferential calculus formulation" of the respective equation. This may be regarded as a generalization of the problem of formulating the equation as a reduction of the selfdual Yang-Mills equation. Indeed, in the case under consideration, it is of great help that an embedding of the non-autonomous chiral model in the (m × m) selfdual Yang-Mills equation is known [35,36,37,38,39,40], and a bidifferential calculus formulation is then obtained from that of the selfdual Yang-Mills equation [1], see Section 4. Once this is at hand, the remaining computations are rather straightforward. Section 6 contains some concluding remarks.

Preliminaries
Basic def initions. A graded algebra is an associative algebra Ω over C with a direct sum decomposition Ω = r≥0 Ω r into a subalgebra A := Ω 0 and A-bimodules Ω r , such that Ω r Ω s ⊆ Ω r+s . A bidifferential calculus (or bidifferential graded algebra) is a unital graded algebra Ω equipped with two (C-linear) graded derivations d,d : Ω → Ω of degree one (hence dΩ r ⊆ Ω r+1 , dΩ r ⊆ Ω r+1 ), with the properties and the graded Leibniz rule for all χ ∈ Ω r and χ ′ ∈ Ω. This means that d andd both satisfy the graded Leibniz rule. In Section 3 we consider a more narrow class of graded algebras. A bidifferential calculus within this class is then specified in Section 4. Dressing a bidif ferential calculus. Let (Ω, d,d) be a bidifferential calculus. Replacing d κ in (2.1) by with a 1-form A (i.e. an element of Ω 1 ), the resulting condition D 2 κ = 0 (for all κ ∈ C) can be expressed as If these equations are equivalent to a PDDE or a system of PDDEs for a set of functions, we say we have a bidifferential calculus formulation for it. This requires that A depends on these functions and the derivations d,d involve differential or difference operators. There are several ways to reduce the two equations (2.2) to a single one. Here we only consider two of them. 1. We can solve the first of (2.2) by setting where α ∈ A is d-constant 2 and β ∈ A isd-constant, and both have to be invertible. Sincē is If P satisfies dP = (dP )P, (2.8) this reduces to

A solution generating method
Let (C N ) denote the exterior (Grassmann) algebra of the vector space C N and Mat(m, n, B) the set of m × n matrices with entries in some unital algebra B. We choose A as the algebra of all finite-dimensional matrices (with entries in B), where the product of two matrices is defined to be zero if the sizes of the two matrices do not match, and assume that Ω = A ⊗ (C N ) is supplied with the structure of a bidifferential calculus. In the following, I = I m and I = I n denote the m × m, respectively n × n, identity matrix.
Proof . Using the last three of (3.1) we obtain Multiplication by U from the left and by V from the right, and usingdI = 0, leads tō dg = U (dX −1 )V g = (dφ)g.
Hence φ and g solve the Miura transformation equation (2.10). We did not use the first of (3.1), but it arises as an integrability condition: 0 =d 2 X = (dX)[(dP )P −dP ]. where the last term vanishes. If P and R are sufficiently independent, this implies that the third of (3.1) is satisfied. In particular, this holds if B is the algebra of complex functions of some variables and if P and R have no eigenvalue in common.
Appendix A explains how Proposition 3.1 arises from a theorem that has been applied in previous work to generate soliton solutions of several integrable PDDEs.

The non-autonomous chiral model
The PDE defining the non-autonomous chiral model can be obtained as a reduction of the self-dual Yang-Mills (sdYM) equation (see e.g. [35,36,37,38,39,40]). In an analogous way, a bidifferential calculus for the non-autonomous chiral model can be derived from a bidifferential calculus for the sdYM equation (also see [41]). In coordinates ρ, z, θ, where ρ > 0, it is given by (4.1) Here e.g. f z denotes the partial derivative of a function f (of the three coordinates) with respect to z, and ζ 1 , ζ 2 is a basis of 1 (C 2 ). d andd extend to matrices of functions and moreover to Ω = A ⊗ (C 2 ) with A = Mat(m, m, C), treating ζ 1 , ζ 2 as constants. The coordinate θ is needed to have the properties of a bidifferential calculus, but we are finally interested in equations for objects that do not depend on it.
A (matrix-valued) function is d-constant (d-constant) iff it is z-independent and only depends on the variables θ, ρ through the combination ρe θ (respectively ρe −θ ). It is d-andd-constant iff it is constant, i.e. independent of z, θ, ρ.
For an m × m matrix-valued function g, (2.4) takes the form with any constant c and θ-independentg, for the latter we obtain the non-autonomous chiral model equation 3 In Section 4.1, we derive a family of exact solutions by application of Proposition 3.1. In Appendix B we recover two familiar linear systems (Lax pairs) for this equation.
whereφ is θ-independent, we obtaiñ which is related to the non-autonomous chiral model by the Miura transformatioñ Symmetries. (4.2) is invariant under each of the following transformations, and thus, more generally, any combination of them.

A family of exact solutions
Let us first consider the equationdP = (dP )P , which is the first of (3.1). Using the above bidifferential calculus, it takes the form and assuming thatP does not depend on θ, this translates tõ The proof of the following result is provided in Appendix C.
Lemma 4.1. The following holds.
(1) IfP and I +P 2 are invertible, the system (4.4) implies with a constant matrix B.
Remark 4.1. IfP is diagonal, then (4.5) becomes the set of quadratic equations (2.11) in [6] (or (1.67) in [7]), which determine the "pole trajectories" in the framework of Belinski and Zakharov. In our approach, there are more solutions sinceP need not be diagonal.  [42]. But this would be unnecessarily restrictive, see Section 4.2.

Remark 4.3. Under the assumption that
also see (C.1). For the bidifferential calculus under consideration,dP = (dP )P is therefore equivalent todP = P dP . The latter is one of our equations for R in Proposition 3.1. Setting withR θ-independent, invertibleP andR both have to solve (4.5).
The third of (3.1) becomes Assuming that U and V are θ-independent, and recalling the θ-dependence of φ, the formula for φ in (3.2) requires X = e θX with θ-independentX. Hencẽ The last of (3.1) becomes the θ-independent Sylvester equatioñ Now Proposition 3.1 implies the following.
Proposition 4.1. Let n × n matricesP andR be solutions of (4.5) (with a matrix B, respectively B ′ ), with the properties that they commute with their derivatives w.r.t. ρ and z, and that I +P 2 and I +R 2 are invertible. Furthermore, let spec(P ) ∩ spec(R) = ∅ andX an invertible solution of the Sylvester equation (4.7) with constant m × n, respectively n × m, matrices U and V . Theñ with any constant invertible m × m matrix 4 g 0 , solves the non-autonomous chiral model equation (4.2).
Proof . As a consequence of the spectrum condition, a solutionX of the Sylvester equation (4.7) exists and is unique. The further assumptions forP andR are those of Lemma 4.1, part (2). Furthermore, (4.6) is a consequence of (4.7) if the spectrum condition holds (also see Remark 3.1). Now our assertion follows from Proposition 3.1 and the preceding calculations.
Remark 4.4. The determinant of (4.8) is obtained via Sylvester's theorem, where we used the Sylvester equation (4.7) and assumed that it has an invertible solution.
Remark 4.5. As an obvious consequence of (4.7), U and V enterg given by (4.8) only modulo an arbitrary scalar factor different from zero. We also note that a transformatioñ with constant invertible n × n matrices T 1 , T 2 , leaves (4.5), (4.6), (4.7) and (4.8) invariant. As a consequence, without restriction of generality, we can assume that the matrix B in (4.5), and the corresponding matrix related toR, both have Jordan normal form.
If they have no eigenvalue in common, then (4.7) has a unique solution given by the Cauchy-like matrixX It remains to solve (4.5) (choosing B diagonal), which yields Since we assume that {p i } ∩ {r i } = ∅, the assumptions of Proposition 4.1 are satisfied. It follows that, with the above data, (4.8) solves the nonautonomous chiral model equation.
The case whereP orR is non-diagonal is exploited in the next subsection. But Example 4.1 will be sufficient to understand most of Section 5.

More about the family of solutions
Introducing matrices A and L via A = (zI + B) 2 + ρ 2 I,P = ρ −1 (L + zI + B), According to Remark 4.5, we can take B in Jordan normal form, Let us first consider the case where B is a single r × r Jordan block, Then we have and thus by use of the generalized binomial expansion formula, noting that M r r = 0 as a consequence of N r r = 0. Hence we obtain the following solution of (4.5), which is an upper triangular Toeplitz matrix. In particular, we havẽ These matrices are obviously nested and, from one to the next, only the entry in the right upper corner is new.
For the above Jordan normal form of B, solutions of (4.5) are now given by 5 P = block-diag(P n 1 , . . . ,P ns ), where the blocks typically involve different constants replacing b, i.e. different eigenvalues of B. SinceP ρ andP z obviously commute withP , and since I+P 2 is invertible 6 for ρ > 0, Lemma 4.1, part (2), ensures thatP solves (4.4). IfP has the above form, andR a similar form, and ifP andR have disjoint spectra, it remains to solve the Sylvester equation 7 (4.7) in order that (4.8) yields solutions of the non-autonomous chiral model equation. This leads to a plethora of exact solutions. We postpone an example to Section 5, where additional conditions considerably reduce the freedom we have here, see Example 5.2.

Reductions of the non-autonomous chiral model to Ernst equations
According to Section 4, a particular involutive symmetry of the non-autonomous chiral mo- (4.2) therefore admits the generalized unitarity reductiong † γg = γ, which means thatg belongs to the unitary group U (m; γ). 8 Another reduction, associated with an involutive symmetry, is g † =g. Imposing both reductions simultaneously, amounts to setting translates these conditions into γP † γ = P, P 2 = P.
In particular, P is a projector. If we require in addition that rank(P) = 1, which for a projector is equivalent to tr(P) = 1 [43, Fact 5.8.1], the following parametrization ofg can be achieved (also see e.g. [44,45,46,47]), where v is an m-component vector with v † γv = 0. This parametrization is invariant under v → cv with a nowhere vanishing function c, so that the first component of v can be set to 1 in the generic case where it is different from zero. If γ has signature m−1, [44,45,46]. The condition tr(P) = 1 corresponds to We also note that det(g) = − det(γ). The following result, which we prove in Appendix C, shows how the reduction conditions (5.1) and (5.3) can be implemented on the family of solutions of the non-autonomous chiral model obtained via Proposition 4.1.
6 Note that Ir +P Under the stated conditions the Sylvester equation possesses a unique solution and a vast literature exists to express it. 8 If γ has p positive and q negative eigenvalues, this is commonly denoted U (p, q). (2) If moreover the relations hold, theng given by (4.8) is Hermitian.
Then we have Γ 2 = I and (5.4) holds. If spec(P ) ∩ spec(R) = ∅, the corresponding Sylvester equation has a unique solutionX. According to part (1) of Proposition 5.1,g given by (4.8) satisfies the reduction conditions (5.6). This is a way to superpose solutions from the class obtained in Section 3, preserving the constraints (5.4). We simply block-diagonally compose the matrix data associated with the constituents. In an obvious way, this method can be extended to part (2) of Proposition 5.1.

Solutions of the Ernst equation of general relativity
We choose m = 2 and with a complex function E and its complex conjugateĒ. Then (5.2) takes the form so that (4.2) now becomes the Ernst equation where e.g. ∂ ρ denotes the partial derivative with respect to ρ. This equation determines solutions of the stationary axially symmetric Einstein vacuum equations. The following statements are easily verified.
with p i , r i given by (4.9). Choosing so that det(g 0 ) = 1, the second of the reduction conditions (5.11) is solved by setting noting that −1/p i is given by the expression for p i with j i exchanged by −j i . We shall write p, r instead of p 1 , r 1 . With U = (u ij ), V = (v ij ), the remaining constraintg 12 =g 21 is solved by 11 In the following we assume that u 11 and v 11 are different from zero and write Then u 11 , u 12 , v 11 , v 21 drop out ofg. Without restriction of generality, we can therefore choose them as u 11 = 1, u 12 = −u, v 11 = 1 and v 21 = −v, hence Then U and V commute with γ.g is real in particular if either of the following conditions is fulfilled.
(1) p, r real (which means b, b ′ real) and u, v real.
By a shift of the origin of the coordinate z, we can arrange in both cases that where b ∈ R in case (1) and b ∈ i R in case (2). Using and introducing we obtain . (5.14) Setting jj ′ = −1, the cases (1) and (2) simply distinguish the non-extreme and the hyperextreme Kerr-NUT space-times (see e.g. [50]). The constants satisfy m 2 + l 2 − a 2 = b 2 .
where p = ρ −1 (z + b + r), r = ± (z + b) 2 + ρ 2 , and r i is also of the form (4.9) with a constant b ′ i = b. The conditions for U and V restrict these matrices to the form with constants u i , v i . If r 1 = r 2 =: r (i.e. b ′ 1 = b ′ 2 ), it turns out thatg does not depend on r and v i , and we obtain with the parameters which satisfy m 2 − a 2 + l 2 = 0. E is the Ernst potential of an extreme Kerr-NUT space-time.

Solutions of the Ernst equations in the Einstein-Maxwell case
See e.g. [6,24,51] for other derivations, and also [52], as well as the references cited there.
(also see [53,47]). We have where ⊺ denotes transposition. In the following we consider the case m = 3, where (4.2) becomes the system of Ernst equations which determine solutions of the stationary axially symmetric Einstein-Maxwell equations (without further matter fields). If E = 1 and Φ = 0, theng reduces to which corresponds to the Minkowski metric.

Example 5.4 (Demiański-Newman). Let n = 2 and
with p, r as in (4.9) with constants b, b ′ . Solving g 0 γU = U Γ and ΓV = V g 0 γ, and recalling that U and V enter the solution formula (4.8) only up to an overall factor, leads to According to Proposition 5.1, part (1), in order to obtain solutions of the Ernst equations it remains to determine conditions under whichg is Hermitian. By explicit evaluation one finds that this is so if one of the following sets of conditions is satisfied.
Without restriction of generality we can set b ′ = −b, so that p and r are given by (5.12). The Ernst potential E is again of the form (5.14), where now The second Ernst potential is given by , (5.19) where In both cases, the parameters a, l, m, q e , q m are real and satisfy Cases (1) and (2) correspond to a non-extreme, respectively hyperextreme, Demiański-Newman space-time (see e.g. [50]). q e and q m are the electric and magnetic charge, respectively. Whereas (2) can be neatly expressed via (5.7), we have been unable so far to find a corresponding formulation for the conditions (1) in terms of the matricesP ,R, U , V , also see Remark 5.4.

Conclusions
We addressed the m × m non-autonomous chiral model in a new way, starting from a very simple and universal solution generating result within the bidifferential calculus approach. This resulted in an infinite family of exact solutions for any matrix size m, parametrized by matrices subject to a Sylvester equation. To solve the latter is a well-studied and fairly simple problem. At least in the compact form presented in this work, according to our knowledge these solutions have not appeared previously in the literature. The non-autonomous chiral model originally appeared in reductions of Einstein's equations. We demonstrated in Section 5 that the "multi-solitons" on a flat background, known in the case of stationarity, axial symmetry and vacuum, respectively electrovacuum, are indeed contained in the family of solutions that we obtained in Section 4 for the non-autonomous chiral model equation. More precisely, we found conditions to be imposed on the matrices that parametrize the latter solutions such that (in the cases m = 2, respectively m = 3) they become solutions of the Ernst equation(s) of general relativity. Only in the case of non-extreme multi-Demiański-Newman solutions we were not (yet) able to find a corresponding characterization of the matrix data.
Beyond the solutions found e.g. by Belinski and Zakharov, which in the present work correspond to diagonal matricesP andR, there are solutions associated with non-diagonal matrix data. It may well be that such solutions can be obtained alternatively e.g. in the Belinski-Zakharov framework with a dressing matrix involving higher order poles, or by taking suitable limits where some poles coincide. In any case, our approach yields these solutions directly. Moreover, relaxing the spectral condition for the matricesP andR, the Sylvester equation has further solutions, provided that the matrix V U on its right hand side is appropriately chosen. This is another possibility to obtain new solutions. Finally, we should mention the possibility to make sense of the limit 14 n → ∞ (where n × n is the size of the matrices that parametrize the solutions). In conclusion, at present it is not quite clear what the generated class of solutions really embraces.
Moreover, using the original method of Belinski and Zakharov, in the Einstein-Maxwell case no appropriate reduction conditions could be found (cf. [54]), and a different approach had to be developed [23,7]. We had less problems in this respect.
On the other hand, the Belinski-Zakharov approach, the modified approach of Alekseev [23] in the Einstein-Maxwell case, and others can also be used to generate "solitons" on a nonflat background. Perhaps a corresponding extension of Proposition 3.1 exists. This is also suggested by the relation with Darboux transformations in Appendix A. In any case, here we have a limitation of Proposition 3.1 (which is not a limitation of the bidifferential calculus framework, which offers various methods [1]), but we have the advantage of a very simple and general result that covers physically interesting cases.
The appearance of a Sylvester equation is a generic feature of the solution generating result formulated in Proposition 3.1 and in Appendix A (also see [1]). Sylvester equations and their simplest solutions, Cauchy-like matrices, frequently appeared in the integrable systems literature. But this is the first time we came across a Sylvester equation involving non-constant matrix data. A particularly nice feature is the fact that solutions can be superposed by simply composing their matrix data into bigger block-diagonal matrices. The corresponding Sylvester equation still has to be solved, but a unique solution exists if we impose a not very restrictive spectral condition on these matrix data.
In Appendix B we recovered two familiar Lax pairs for the non-autonomous chiral model from the general linear equation (2.7) in the bidifferential calculus framework. Our way toward exact solutions in Section 4 is more closely related to Maison's Lax pair than to that of Belinski and Zakharov. We eliminated the θ-dependence, whereas in the Lax pair of Belinski and Zakharov the θ-dependence is kept and it involves derivatives with respect to this "spectral parameter".
Our results extend beyond the Einstein-Maxwell case and are also applicable to higherdimensional gravity theories (see e.g. [9,10,11,12,13,14,15,16,17,18,19]). Besides that, other reductions of the non-autonomous chiral model (for some m) are of interest, see e.g. [20,21], and the set of solutions that we obtained in this work will typically be reducible to solutions of them.
Since 3), respectively (2.4), from solutions of linear equations. However, the equations for P and R arise as integrability conditions of the latter. In previous work [1,32,33,34], we chose P and R as d-and d-constant matrices, which indeed reduces the equations that have to be solved to only linear ones, and we recovered (and somewhat generalized) known soliton solution families. In case of the non-autonomous chiral model and, more specifically, its reduction to the Ernst equation, it turned out to be necessary to go beyond this level, and thus to consider genuine solutions of the nonlinear equations for P and R, in order to obtain relevant solutions like those associated with multi-Kerr-NUT space-times and their (electrically and magnetically) charged generalizations. This also suggests a corresponding application of the theorem (or Proposition 3.1) to other integrable PDDEs.
A Via a Darboux transformation and a projection to a non-iterative solution generating result Lemma A.1. Let P be invertible. The transformation where X is an invertible solution of (2.7) with A = dφ = (dg)g −1 , anddP = (dP )P , maps a solution of the Miura transformation equation (2.10) into another solution.
(A.1) is an essential part of a Darboux transformation, cf. [1]. In the following we will use this result to derive a theorem which essentially reduces to Proposition 3.1, see Remark A.2. Proof . Since (φ, g) is assumed to solve (2.10), we havē Multiplying by U from the left and by V from the right, we obtain which is equivalent to (2.10). In addition we require that where U ∈ Mat(m, n, B) and V ∈ Mat(n, m, B) are dandd-constant, and Y ∈ Mat(n, n, B). Then also solve the Miura transformation equation (2.10), and thus (2.3), respectively (2.4).
Proof . Since we assume that (−R, S) solves the Miura transformation equation (2.10) in Mat(n, n, B), according to Lemma A.1 this also holds for the pair Using the first of (A.6), we find that (A.2) holds withφ = Y X −1 . Now (A.3) yields the asserted formulas for φ and g. According to Lemma A.2, φ and g solve the Miura transformation equation (2.10). Together with (A.5), the first of (A.6) implies which is satisfied if the last two conditions of (A.6) hold.
Remark A.1. This theorem generalizes a previous result in [1], which has been applied in [1,32,33,34] with d-andd-constant P , R, in which case only linear equations have to be solved in order to generate solutions of (2.3), respectively (2.4).
The above derivation shows that the theorem may be regarded as a combination of a Darboux transformation (Lemma A.1), on the level of matrices of arbitrary size, and a projection mechanism (Lemma A.2). The projection idea can be traced back to work of Marchenko [57]. More generally, the above result can be formulated in terms of suitable operators, replacing the matrices that involve a size n.
The next remark shows that, with mild additional assumptions, Theorem A.1 reduces to Proposition 3.1.
with any invertible Q ∈ Mat(n, n, B), leaves the expressions for φ and g in (A.7) invariant. This is then also a symmetry transformation of (A.5) and (A.6) ifdQ −1 = (dQ −1 )P . As a consequence, under the assumptions that Y is invertible, without restriction of generality we can set Y = I, where I is the n × n identity matrix. Then φ is given by the expression in (3.2). We further note that (A.4) and the second of (A.6) implyd(RS) = 0. Assuming that R is invertible, we thus have S = R −1 C with an invertibled-constant C. The expression for g in (A.7) now takes the form assuming temporarily invertibility of U C −1 V . Together with φ, this remains a solution of (2.10) if we drop the last factor, so that This expression also makes sense without the above additional invertibility assumptions. We can still translate it into a simpler form. From the first of (A.6), which now has the form of the last of (3.1), we obtain Multiplication by U from the left and by V from the right, and use in our last formula for g, leads to the expression for g in (3.2).
B Linear systems for the non-autonomous chiral model
Choosing P = pI with a function p(ρ, z), the equations forP can easily be integrated, which results in 15 where b is an arbitrary constant. In terms of X =g −1X the above linear system, simplified by setting c 1 = c 2 = 0, then takes the form X ρ = − p 1 + p 2 g −1g z + pg −1g ρ X ,X z = p 1 + p 2 g −1g ρ − pg −1g z X .
This system is equivalent to a linear system for the non-autonomous chiral model, first found by Maison in 1979 [4] (also see [41]).

B.2 The Belinski-Zakharov Lax pair
Using instead of θ the variable λ = −ρe θ , We consider the linear system (2.7) with P = I, which trivially solves (2.8), i.e. dX = AX + dX. Writing the integrability condition (2.9) takes the form assuming that A, B are λ-independent. Solving the first (zero curvature) condition by the second equation becomes the non-autonomous chiral model equation ρg ρ g −1 ρ + ρg z g −1 z = 0.
The above linear equation leads to This is the Belinski-Zakharov Lax pair [6] (also see [7,Chapter 8]). We note that the "spectral parameter" λ has its origin in a coordinate of the self-dual Yang-Mills equation. We also note that A = (dg)g −1 (using g λ = 0).

C.2 Proof of Proposition 5.1
Using ( Expanding the left hand side and using the Sylvester equation (4.7) to eliminate V U , this indeed turns out to be satisfied. To complete the proof of (1), it remains to derive the trace formula. Using In order to prove (2), we consider the Hermitian conjugate of the Sylvester equation (4.7). By use of (5.7), and with the help of g 0 γU = U Γ, ΓV = V g 0 γ and (g 0 γ) 2 = I, it takes the form By comparison with the original Sylvester equation, the spectrum condition allows us to conclude thatX † = −ΓXΓ.
It follows that U (RX) −1 V g 0 is Hermitian, and thus alsog given by (4.8).