Orbital Linearization of Smooth Completely Integrable Vector Fields

The main purpose of this paper is to prove the smooth local orbital linearization theorem for smooth vector fields which admit a complete set of first integrals near a nondegenerate singular point. The main tools used in the proof of this theorem are the formal orbital linearization theorem for formal integrable vector fields, the blowing-up method, and the Sternberg-Chen isomorphism theorem for formally-equivalent smooth hyperbolic vector fields.


Introduction
The main purpose of this paper is to show the following orbital linearization theorem for smooth (C ∞ ) vector fields which admit a complete set of first integrals near a nondegenerate singular point: Theorem 1.1. Let X be a smooth vector field in a neighborhood of O = (0, . . . , 0) in R n , which vanishes at O and satisfies the following conditions: i) (Complete integrability): X admits n − 1 functionally independent smooth first integrals F 1 , . . . , F n−1 , i.e. X(F 1 ) = . . . = X(F n−1 ) = 0 and dF 1 ∧ . . . ∧ dF n−1 = 0 almost everywhere. ii) (Nondegeneracy 1): The semisimple part of the linear part of X at O is non-zero, and the ∞-jets of F 1 , . . . , F n−1 at O are funtionally independent. iii) (Nondegeneracy 2): If moreover 0 is an eigenvalue of X at O with multiplicity k ≥ 1, then the differentials of the functions F 1 , . . . , F k are linearly independent at O: dF 1 (O) ∧ . . . ∧ dF k (O) = 0. Then there exists a local smooth coordinate system (x 1 , . . . , x n ) in which X can be written as where X (1) is a semisimple linear vector field in (x 1 , . . . , x n ), and F is a smooth first integral of X (1) , i.e. X (1) (F ) = 0, with F (O) = 1.
The above theorem is in fact more than mere orbital linearization: not only that X is orbitally equivalent to its linear part X (1) , but also the factor F in the expression X = F X (1) in a normalized coordinate system is a first integral of X and X (1) . In [9], this kind of linearization is called geometrical linearization.
The formal and analytic case of the above theorem also holds and was shown in [9] in a more general context of integrable non-Hamiltonian systems of type (p, q), i.e. with p commuting vector fields and q common first integrals, where p + q = n is the dimension of the manifold. The vector fields that we study in this paper are integrable of type (1, n − 1), i.e. just one vector field and n − 1 first integrals.
The nondegeneracy condition in Theorem 1.1 is a bit stronger than the nondegeneracy condition in [9]: in [9] the (formal or analytic) vector field X is called integrable nondegenerate if it satisifes the above conditions i) and ii), without the need of condition iii). In fact, in the formal and analytic case, condition iii) is a simple consequence of the first two conditions and the theorem about the existence of (formal or analytic) Poincaré-Dulac normalization [8,9]. However, in the smooth case, we don't have a proof of the fact that condition iii) follows from conditions i) and ii) in general, though we do have a proof of this fact for dimension 2.
The rest of this paper is organized as follows. Section 2 is devoted to some preliminary results, including the classification of the nondegenerate singularities of completely integrable vector fields into (strong/weak) elliptic and hyperbolic cases (Lemma 2.2), and the normalization up to a flat term (Proposition 2.3). These preliminary results are used in the proof of Theorem 1.1 which is presented in Section 3. Finally, in Section 4, we show that, at least in the case n = 2, condition iii) Theorem 1.1 is a consequence of the first two conditions, and can be dropped from the formulation of the theorem (Theorem 4.1). We conjecture that condition iii) is redundant in the higher-dimensional case as well.
This paper is part of our program of systematic study of the geometry and topology of integrable non-Hamiltonian systems. In particular, Theorem 4.1, which is a refinement of Theorem 1.1 in the case of dimension 2, is the starting point of our joint work with Nguyen Van Minh on the local and global smooth invariants of integrable dynamical systems on 2-dimensional surfaces [10].

Preliminary results
2.1. Adapted first integrals. We have the following simple lemma, which is similar to the well-known Ziglin's lemma [7] Lemma 2.1. Let G 1 , . . . , G m be m formal series in n variables which are functionally independent. Then there exists m polynomial functions of m variables P 1 , . . . , P m such that the homogeneous (i.e. lowest degree) parts of the formal series of P 1 (G 1 , . . . , G m ), . . . , P 1 (G 1 , . . . , G m ) are functionally independent.
The proof of the above lemma follows exactly the same lines as the proof of Ziglin of his lemma in [7], and our situation is simpler than the situation of meromorphic functions considered by Ziglin.
Let X be a smooth completely integrable vector field with a singularity at O. We will say that the smooth first integrals F 1 , . . . , F n−1 of X are adapted first integrals if where denotes the homogeneous part (consisting of non-constant terms of lowest degree in the Taylor expansion) of F i at O. Using the above lemma to replace the first integrals F 1 , . . . , F n−1 of X by appropriate polynomial functions of them if necessary, from now on we can assume that F 1 , . . . , F n−1 are adapted.

2.2.
The eigenvalues of X. The fact that X admits n − 1 first integrals implies that the X is very resonant at O. More precisely, we have: Let (X, F 1 , . . . , F n−1 ) be smooth nondegenerate at O, i.e. they satisfy the conditions of Theorem 1.1. Then the linear part of X at O is semisimple, and there is a positive number λ > 0 such that either all the eigenvalues of X at O belong to λZ, or all of them belong to √ −1λZ.
Proof. We can assume that H 1 , . . . , H n−1 are functionally independent, where H i denotes the homogeneous part of F i . The equality X(F i ) = 0 implies that where X (1) is the linear part of X, and X ss is the semisimple part of X (1) in the Jordan-Dunford decomposition. We can write in a complex coordinate system. Recall that the ring of polynomial first integrals of n i=1 λ j z j ∂ ∂z i is generated by the monomial functions n i=1 z a i i which satisfies the resonance relation The fact that H 1 , . . . , H n−1 are independent implies that Equation (2.4) has n − 1 linearly independent solutions which belong to Z n + , which in turn implies that there is a complex number λ such that λ 1 , . . . , λ n ∈ λZ. Remark that if the spectrum of X ss contains a complex eigenvalue λ 1 ∈ C \ (R ∪ √ −1R), then its complex conjugate λ 1 is also in the spectrum because X is real, and λ 1 and λ 1 cannot belong to λZ at the same time for any λ. Thus any eigenvalue of X ss is either real or pure imaginary. If there is one real non-zero eigenvalue, then we can choose λ ∈ R + , otherwise we can choose λ ∈ √ −1R + . Notice that λ = 0 because at least one eigenvalue of X ss is non-zero by our assumptions.
The common level sets of H 1 , . . . , H n−1 are 1-dimensional almost everywhere, and since both X (1) and X ss are tangent to these common level sets, we have that X (1) ∧ X ss = 0, which implies that X (1) is semisimple, i.e. X (1) = X ss .
With the above lemma, we can divide the problem into 4 cases (here I. Strongly hyperbolic (or hyperbolic without eigenvalue 0): λ i ∈ λR * ∀i. II. Weakly hyerbolic (or hyperbolic with eigenvalue 0): III. Strongly elliptic (or elliptic without eigenvalue 0): 2.3. Linearization up to a flat term. Using the geometric linearization theorem of [9] in the formal case, we get the following proposition: Proposition 2.3 (Linearization up to a flat term). Assume that X satisfies the hypotheses of Theorem 1.1. Then there is a local smooth coordinate system (x 1 , . . . , x n ) in which X can be written as is the linear part of X in the coordinate system (x 1 , . . . , x n ), F is a smooth first integral of X (1) , and f lat means a smooth term which is flat at O.

2.4.
Reduction to the case without eigenvalue 0. Assume that X has zero eigenvalue at O with multiplicity k, and dF 1 ∧. . .∧F k = 0, i.e. we can use F 1 , . . . , F k as the first k coordinates in our local coordinate systems. Since the vector field X preserves x 1 , . . . , x k , we can view it as a k-dmiensional family of vector fields on (n − k)-dimensional spaces We will always assume that the vector field X satisfies the hypotheses of Theorem 1.1. The fact that the linear part of X is semisimple is established by Lemma 2.2. In view of Subsection 2.4, it suffices to prove Theorem 1.1 for the cases without zero eigenvalue, by a proof whose parametrized version also works the same.
3.1. The hyperbolic case. Assume that X is hyperbolic without eigenvalue 0. According to Proposition 2.3, we can write X = Y + f lat, where Y = F X (1) is a smooth hyperbolic integrable vector field in normal form. Since X and Y are hyperbolic and coincide up to a flat term, Sternberg-Chern theorem [6,2] says that X is locally smoothly isomorphic to Y , i.e. there is a smooth coordinate system in which X can be written as X = F X (1) , where F is a smooth first integral of X (1) . Theorem 1.1 is proved in the hyperbolic case without eigenvalue 0.
3.2. The elliptic case. In this subsection, we will assume that all the eigenvalues of X at O are non-zero pure imaginary. Using Proposition 2.3, we can assume that X = F X (1) + f lat in a local smooth coordinate system (x 1 , . . . , x n ), where F is a smooth function such that F (O) = 1. Put Y = X/F. Then Y has the same first integrals as X, and The fact that X is of strong elliptic type implies immediately that the dimension n is even, the eigenvalues of X at O are ± √ −1a 1 , . . . , ± √ −1a n/2 where a 1 , . . . , a n/2 are positive real numbers, and we can choose the coordinates (x 1 , . . . , x n ) such that ).
According to Lemma 2.2, we can choose λ > 0 such that a 1 /λ, . . . , a n/2 /λ are natural numbers whose greatest common divisor is 1. Proof. The vector field (dF 1 ∧ . . . ∧ dF n−1 ) ( ∂ ∂x 1 ∧ . . . ∧ ∂ ∂xn ) is tangent to Y , and therefore it is divisible by Y , i.e. we can write where G is a smoth non-flat function at O. Notice that the singular locus of the map (F 1 , . . . , F n−1 ) : It is clear that, by continuity, the set of all points x ∈ U such that the orbit of Y through x is periodic of period ≤ 3π/λ is a closed subset of U . We want to show that this set is actually equal to U (provided that U is small enough). Consider the singular locus Since G is non-flat at O, we can choose a coordinate system (z 1 , . . . , z n ) which is a linear transformation of the coordinate system (x 1 , . . . , x n ), such that the homogeneous part G (h) of G has the form does not vanish in U , by the classical Rolle's theorem on each line {z 2 = const, . . . , z n = const} in U there are at most k zeros of the function G, the singular locus S is of dimension at most n − 1, and the function G is not flat at any point of S. Take a point q ǫ = (z 1 = ǫ, z 2 = 0, . . . , z n = 0) ∈ U with ǫ > 0 small enough. Take the (n − 1)-dimensional ball B n−1 (q ǫ , ǫ K ) of radius ǫ K which is orthogonal to the vector Y (q ǫ ) at the point q ǫ in the coordinate system (z 1 , . . . , z n ), for a certain positive number K to be chosen below. Denote by φ the Poincaré map of the flow of Y on D. A-priori this map does not necessarily fixes the point q ǫ . But due to the fact that Y = X (1) + f lat and the flow of X (1) is periodic, the distance from q ǫ to φ(q ǫ ) is smaller than ǫ K+1 (provided that ǫ is small enough). We can choose K large enough so that the restriction of the map (F 1 , . . . , F n−1 ) to B n−1 (q ǫ , ǫ K ) is injective. Due to the invariance of the functions F i with respect to the vector field Y , the Poincaré map φ, we also have the points q ǫ and φ(q ǫ ) have the same image under the map (F 1 , . . . , F n−1 ). But this map is injective on the ball B n−1 (q ǫ , ǫ K ) which contains these two point, so in fact these two points must coincide, i.e. we have q ǫ = φ(q ǫ ), and the orbit of the flow of Y through the point q ǫ is a periodic orbit, and the period of this orbit is equal to 2π/λ plus a small error term which tends to 0 faster than anu power of ǫ when ǫ tends to 0.
Denote by V the path-connected component of U \ S which contains the points q ǫ . Then the orbit of Y through any point q ∈ V is also periodic and its period is close to 2π/λ (the difference between the period and 2π/λ tends to 0 uniformly when the radius of U tends to 0). This fact can be proved easily by showing that the set of points of V which satisfies the mentioned property is closed and open in V at the same time: closed due to the continuity, and open because (F 1 , . . . , F n−1 ) is regular in V and is preserved by the flow of Y .
Let q ∈ S be a point in the locus S which also lies on the boundary of V . Then by continuity, there is also a number T near 2π/λ such that the time-T flow of Y fixes the point q. In other words, the orbit of Y through q is also periodic, and the period is equal to T or a fraction T /m of T for some natural number m. As before, consider a (n−1)-dimensional ball B n−1 (q, δ) which is centered at q and orthogonal to Y (q), for some δ > 0 small enough. Consider the Poincaré map φ of Y on B n−1 (q, δ) corresponding to the time T (i.e. if the period of the orbit through q is T /m then consider the m-time itaration of the usual Poincaré map). Since the intersection of B n−1 (q, δ) with V contains an open subset of B n−1 (q, δ) whose closure contains q, and the Poincaré map is identity on that open subset by the above considerations, the Poincaré map on B n−1 (q, δ) is equal to the identity map plus a flat term at q. On the other hand, this Poincaré map must preserve the map (F 1 , . . . , F n−1 )| B n−1 (q,δ) , and the determinant of the differential of this map is not flat at q. It implies that the Poincaré map must be identity in a small neighborhood of q in B n−1 (q, δ). Thus, we can "engulf" the set of points shown to have periodic orbits from V to a larger open subset of U which contains the boundary of V . Continuing this engulfing process, we get that the set of points in U having periodic orbits is actually the whole U .
We will linearize Y = X/F orbitally, and then deduce the normalization of X from this linearization. In order to do that, let us consider the blow-up of R n at O, which will be denoted by (3.6) p where U ∋ O is a neigborhood of O in R n and p −1 (O) ∼ = RP n−1 is the exceptional divisor of the blow-up in E. We will need the following simple lemma, whose proof is steaightforward: Denote byG (resp.Z) the pull-back of a function G (resp. vector field Z) via the projection map π : E → U of the blow-up. Then we have (3.7)Ỹ =X (1) +Z in E, whereZ is vector field which is flat along p −1 (O), andX (1) is a smooth periodic vector field in E of period 2π/λ. By Lemma 3.1, the orbits ofỸ are closed, with periods close to the period ofX (1) . Due to the flatness of Z along p −1 (O), the period ofỸ at the points in E is equal to 2π/λ plus a smooth function on E which is flat along p −1 (O). ProjectingỸ back to U and using Lemma 3.2, we get a smooth period function P = 2π/λ + f lat (which is invarant on the orbits) such that P Y is periodic of period 1. In other words, P Y generates a smooth T 1 -action. Using the classical Cartan-Bochner smooth linearization theorem for compact group actions, we find a smooth coordinate system, which we will denote again by (x 1 , . . . , x n ), in which P X/F = P Y is a linear vector field, i.e. in which we have where G is a smooth function and X (1) is a linear vector field which satisfies Formula (3.2). A-priori, the function F given by Proposition 2.3 is not a first integral of X (though it is a first integral of the linear part of X in some coordinate system), and so the function G = 2πF/P λ in Formula (3.8) is not a first integral of X in either. But we can normalize further in order to change G into a first integral. Indeed, by the arguments presented above, we can assume that G is a smooth first integral of X plus a flat term, or we can write G = G 1 (1 + f lat), where G 1 is a first integral of X. Normalizing the new vector field Y = X/G 1 instead of the old Y = X/F, we get a new smooth coordinate system in which P Y = (2π/λ)X (1) , where P is the period function of the new vector field Y , and it is a smooth first integral of the type constant + f lat. In this new coordinate system we have that X is equal to its linear part times a first integral, and Theorem 1.1 is proved in the elliptic case i.e. without eigenvalue 0.
Since our proof for the strong hyperbolic case and the strong elliptic case also works for smooth families of integrable vector fields, Theorem 1.1 is proved.
Remark 3.3. According to a theorem of Schwarz [5], the smooth first integral F in the normal form in the elliptic case can also be written as

The case of dimension 2
The aim of this section is to show that condition iii) in Theorem 1.1 is redundant at least in the case of dimension 2. More precisely, we have: Theorem 4.1. Let X be a smooth vector field in a neighborhood of O = (0, 0) in R 2 , which vanishes at O and satisfies the following conditions: i) (Complete integrability): X admits a smooth first integral F 1 . ii) (Nondegeneracy): The semisimple part of the linear part of X at O is non-zero, and the ∞-jet of F 1 at O is non-constant. Then there exists a local smooth coordinate system (x, y) in which X can be written as where X (1) is a semisimple linear vector field in (x, y), and F is a smooth first integral of X (1) .
Proof. Remark that, in the case of dimension 2, there are only 3 possibilities: elliptic without zero eigenvalue, hyperbolic without zero eigenvalue, and hyperbolic with zero eigenvalue. The first two possibilities are covered by Theorem 1.1. It remains to prove Theorem 4.1 for the case when X has one eigenvalue equal to 0. By Proposition 2.3, we can assume that in a smooth coordinate system (x, y), where f lat 1 and f lat 2 are two flat functions, and F (0) = 0. Denote by the singular locus of X near O, where U denotes a small neighborhood of O in R 2 . The main point is to prove that S is a smooth curve. If S is a smooth curve, then we can write S = {y = 0}, the vector field X is divisible by x, i.e. Y = X/x is still a smooth vector field, which is non-zero at O, and therefore locally rectifiable and admits a first integral G such that dG(0) = 0. But G is also a first integral of X, so condition iii) of Theorem 1.1 is also satisfied, and Theorem 4.1 is reduced to a particular case of Theorem 1.1. Denote by the set of points where the ∂ ∂x -component of X vanishes. It is clear that S ⊂ S 1 , and S 1 is a smooth curve tangent to the line {x = 0} at O by the inverse function theorem. We will show that S = S 1 .
Clearly, S 1 ⊂ C (provided that U is small enough). The non-flat first integral F 1 of X in the coordinate system (x, y) has the type (4.6) is a non-flat smooth function. It implis that the level sets of F 1 in the cone C are smooth curves which are nearly tangent to the lines {y = const}. In particular, each level set of F 1 in C intersects with S 1 at exactly 1 point. Since X is tangent to these level sets, and the ∂ ∂x -component of X vanishes at the intersection points of these level sets with S 1 , it follows that X itself vanishes at these intersetion points. But every point of S 1 is an intersection point of S 1 with a level set of F 1 . Thus X vanishes on S 1 , and we have S = S 1 .
Remark 4.2. Two-dimensional elliptic-like vector fields, i.e. those vector fields whose orbits near a singular point are closed, are also called centers in the literature. There is a recent interesting theorem of Maksymenko [4] about the orbital linearization of the center, without the assumption on the existence of a first integral, but with an assumption on the periods of the periodic orbits. Maksymenko's theorem is similar to and a bit stronger than the elliptic case of Theorem 4.1 because his assumptions are weaker, and the conclusions are the same. His proof is also based on the formal normalization and the blowing-up method.
Remark 4.3. Some of the arguments of the proof of Theorem 4.1 are still valid in the n-dimensional case where 0 is an eigenvalue with multiplicity k ≥ 1. In particular, one can still show that, even without condition iii) of Theorem 1.1, the local singular locus of X is still a smooth k-dimensional manifold. However, it is more difficult to show that there is still a local regular invariant (n−k)-dimensional foliation. If one can show the existence of this regular regular invariant foliation, then one can drop condition iii) from the statement of Theorem 1.1 because it is a consequence of the first two conditions. Maybe it is possible to use the techniques of Belitskii-Kopanskii [1] together with a kind of desingularization of the first integrals in order to show the existence of an invariant regular foliation, but we don't have a proof so far.