Higher Haantjes Brackets and Integrability

We propose a new, infinite class of brackets generalizing the Fr\"olicher--Nijenhuis bracket. This class can be reduced to a family of generalized Nijenhuis torsions recently introduced. In particular, the Haantjes bracket, the first example of our construction, is relevant in the characterization of Haantjes moduli of operators. We shall also prove that the vanishing of a higher-level Nijenhuis torsion of a given operator is a sufficient condition for the integrability of its generalized eigen-distributions. This result (which does not require any knowledge of the spectral properties of the operator) generalizes the celebrated Haantjes theorem. The same vanishing condition also guarantees that the operator can be written, in a local chart, in a block-diagonal form.


Introduction
In the last two decades, the study of the geometry of Nijenhuis and Haantjes tensors has experienced a resurgence of interest. The notion of Nijenhuis torsion was introduced in [19], [20] by A. Nijenhuis in his study of the integrability of eigen-distributions of operator fields with pointwise distinct eigenvalues.
In [10], the graded bracket nowadays called the Frölicher-Nijenhuis bracket was defined. This bracket is relevant in several geometric contexts, in particular in the theory of almost-complex structures, as clarified by the Newlander-Nirenberg theorem [18], [12]. Slightly before, in the seminal paper [11], J. Haantjes proposed the fundamental notion of torsion bearing his name. In particular, he proved that the vanishing of the Haantjes torsion of a (1, 1) tensor field is a necessary condition for the existence of an integrable frame of generalized eigenvectors. This condition is also sufficient in the case of pointwise semisimple operators.
Recently, new and conspicuous applications of Nijenhuis and Haantjes tensors have been found, for instance, in the characterization of integrable chains of partial differential equations of hydrodynamic type (see e.g. [9], [4]) and in the study of infinite-dimensional integrable systems (in particular in connection with the celebrated WDVV equations of associativity and the theory of Dubrovin-Frobenius manifolds [15]- [17]). In [22]- [25] we have proposed the notion of Haantjes algebras and the related ones of ωH and P H manifolds as the natural setting for the formulation of the theory of classical finite-dimensional Hamiltonian integrable systems.
The aim of this work is twofold. Our first goal is to introduce a new, infinite class of brackets that generalize the Frölicher-Nijenhuis bracket. The first representative of our class coincides with it; the second one is already a new example, the Haantjes bracket H A,B (X, Y ); by means of a recursive procedure we also define infinitely many novel higher-level brackets. Here "bracket" means that the tensors introduced depend on a pair (A, B) of (1,1) tensor fields.
A simple reduction of this family, obtained when each representative depends on a pair (A, A) of copies of the same operator field, coincides (up to a constant) with the family of generalized torsions defined independently (and from a different perspective) by Y. Kosmann-Schwarzbach in [13] and by ourselves in an early, preprint version of [23] of 2017.
By means a new "tower" of brackets, we aim to study the geometry of very general families of operators, as the triangularizable ones, that (except in very specific cases) have non-vanishing Haantjes torsion.
We have ascertained the geometric relevance of our higher brackets in several important situations. Precisely, as stated in Theorem 23, given two commutative semisimple operators, they generate a commutative Haantjes module [23] if and only if their Haantjes bracket vanishes. Also, we study several further algebraic properties of this new bracket.
Our second goal is to clarify the geometric meaning of the "generalized Nijenhuis torsions" of higher level introduced in [13], [23]. In Section 5 (Proposition 26 and Corollary 27) we prove that the vanishing of the generalized Nijenhuis torsion τ (n−1) A (X, Y ) = 0 of level (n − 1) of a nilcyclic (i.e. both nilpotent and cyclic) operator field A on a manifold of dimension n is necessary for the existence of a local chart where A takes a triangular form (see Eq. (46)).
The main theorem of the present work, Theorem 40 of Section 6, concerns the integrability properties of the generalized eigen-distributions (i.e., distributions of generalized eigenvector fields) of an operator field. A seminal result, due to Haantjes [11], states that in the case of a semisimple operator field, a necessary and sufficient condition for the Frobenius integrability of its eigen-distributions of constant rank is that its Haantjes tensor identically vanishes. However, in the general case of a non-semisimple operator, the previous condition is only sufficient. Thus, for the infinite class of operators whose Haantjes tensor is not vanishing, no conclusion can be drawn about integrability of their eigen-distributions.
Our main theorem fills this gap. Indeed, we shall prove that the vanishing of a generalized Nijenhuis torsion τ (m) A (X, Y ) of level m for some integer m ≥ 1 provides us with a sufficient condition for the integrability of the generalized eigendistributions of a given operator field A. In addition, it ensures the integrability of all of their direct sums. Thus, we are able to construct a tensorial test for Frobenius integrability of a very large class of operator fields, which significantly extends the applicability of the original Haantjes torsion criterion.
The interest of our result, in the spirit of Haantjes's theorem, relies crucially on the fact that, in order to ascertain the integrability properties of a given operator, no knowledge a priori of the spectrum of this operator nor of its eigen-distributions is required.
An important consequence of the main Theorem (see Proposition 43) is the fact that an operator with a vanishing generalized Nijenhuis torsion (for some m ≥ 1) admits a local coordinate chart where it takes a block-diagonal form.
In short, the body of results proposed indicates that all of the infinitely many higher level tensors introduced possess a geometric meaning and are relevant in applicative contexts.
An important open problem we address in Section 5 is to decompose a generic operator field in a local coordinate chart as the sum of a diagonal operator and another operator, whose generalized Nijenhuis torsion (of a suitable level) vanishes.
We also believe that the theory of higher brackets proposed in this work could play a significant role, more generally, in the theory of integrable systems, for instance in the study of generic hydrodynamic-type systems, not possessing Riemann invariants. For instance, a potentially interesting area is the study of equations of hydrodynamic type in 2+1 dimensions, namely, equations of the form u t = A(u)u x + B(u)u y , where A(u) and B(u) are operator fields which not necessarily commute [8]. It would be interesting, for instance, to classify the pairs of operators (A(u), B(u)) relevant in the theory of hydrodynamic-type systems by means of suitable tensor conditions ensuring integrability (some preliminary results are presented in Section 3.3.1.).

Preliminaries on the Nijenhuis and Haantjes geometry
In this section, we shall review some basic notions concerning the geometry of Nijenhuis or Haantjes torsions, following the original papers [11,19,10]. Here we shall focus only on the aspects of the theory which are relevant for the subsequent discussion.
Let M be a differentiable manifold, X(M ) the Lie algebra of all smooth vector fields on M and A : X(M ) → X(M ) be a smooth (1, 1) tensor field (namely, an operator field). For the sake of simplicity, the expressions "tensor fields" and "operator fields" will be abbreviated to tensors and operators. In the following, all tensors will be considered to be smooth.
where X, Y ∈ X(M ) and [ , ] denotes the commutator of two vector fields.
A simple, relevant case of Haantjes operator is that of a tensor A which takes a diagonal form in a local chart x = (x 1 , . . . , x n ): are the fields forming the so called natural frame associated with the local chart (x 1 , . . . , x n ). As is well known, the Haantjes torsion of the diagonal operator (3) vanishes.
We also recall that two frames {X 1 , . . . , X n } and {Y 1 , . . . , Y n } are said to be equivalent if n nowhere vanishing smooth functions f i exist, such that

Definition 4. [1]
An integrable frame is a reference frame equivalent to a natural frame.
Remark 5. We wish to point out that the adjectives "diagonalizable" and "semisimple" are both used in the literature, sometimes interchangeably. From now on, we shall call diagonalizable an operator which takes a diagonal form in a natural reference frame (as in formula (3)), whereas we shall say that an operator is pointwise semisimple (or semisimple tout court ) if it admits a local reference frame (not necessarily natural, nor integrable) in which it takes a diagonal form. Diagonalizable operators are obviously semisimple; the converse statement is not true in general. Historically, the problem addressed by Nijenhuis and Haantjes was to ascertain whether a local reference frame constructed out of the eigenvectors of an operator is integrable or not.
It is interesting to observe that the algebraic properties of Haantjes operators are different, and sometimes richer that those of Nijenhuis operators. One useful result is the following (hereafter, I : X(M ) → X(M ) will denote the identity operator). Proposition 6. [2]. Let A be a (1,1) tensor. The following identity holds where f, g : M → R are C ∞ (M ) functions.
Interestingly enough, such a simple property does not hold in the case of a Nijenhuis operator.
Many more examples of Haantjes operators, relevant in classical mechanics and in Riemannian geometry can be found for instance in [21]- [25].

Haantjes brackets
Let M be a differentiable manifold and A, B : X(M ) → X(M ) be two operators.
[10] The Frölicher-Nijenhuis bracket of A and B is the vector-valued 2-form given by 1 The local expression of the components of the Frölicher-Nijenhuis bracket reads . This bracket has relevant geometric applications [18], in particular in the theory of almost-complex structures and in the detection of obstructions to integrability [12]. The bracket is symmetric and R-linear (but not C ∞ (M )-linear) in A and B. In fact, it satisfies the identity Here T(A, B) : X * (M ) × X(M ) × X(M ) → X(M ) is the vector-valued 3-tensor defined by We shall denote by T T (α, X, Y ) := T(α, Y, X) the transposed of T w.r.t. the last two arguments. We also recall that for each operator A, B, and for all α ∈ X * (M ), Note that for each operator B : X(M ) → X(M ), we have Choosing A = B in Eq. (5), one gets twice the Nijenhuis torsion: 1 For sake of clarity, in this article we have renounced to the usual unified notation [· , ·] which, depending on the context, should stands for both the standard Lie bracket of vector fields and the Frölicher-Nijenhuis bracket of operators. Instead, we have preferred to maintain the symbol · , · for the Frölicher-Nijenhuis bracket and to introduce the notation [· , ·] for the Lie bracket of two vector fields and the commutator of two operators.
For all f, g ∈ C ∞ (M ), the following identity holds: This identity allows us to characterize modules of Nijenhuis operators. Examples.

i) Let us consider the couple of Nijenhuis operators
They satisfy conditions (11)-(13); then, they generate a module of Nijenhuis operators.

ii) Let us consider the couple of Nijenhuis operators
∂x k ⊗ dx k , whose Frölicher-Nijenhuis bracket vanishes. In fact, they generate a vector space of Nijenhuis operators.
Inspired by the previous construction, we shall introduce a novel "tower" of higherlevel brackets. The first step is to generalize the Frölicher-Nijenhuis bracket.

3.2.
A new family of higher brackets. Hereafter, we shall present the main algebraic construction of this work, namely the recursive definition of an infinite class of new brackets of couples of operators.
None of these brackets for m ≥ 2 is R-linear in A and B; however, they are symmetric in the interchange of A and B.
The following statement can be useful for computational purposes.
Proof. This formula comes directly from the expression in local coordinates of the Frölicher-Nijenhuis bracket (6), applied to the recursive formula (14).
If we take A = B, then the previous family of brackets reduces to the generalized torsions proposed independently in [13] and [23]. Here we remind the main definition of that construction, since it will be crucial in the subsequent discussion.
Here the notation τ We also remind a useful formula, proved in [13] (Section 4.6), by means of a suitable polynomial representation of (1, 2) tensors: Alternatively, this formula can also be proved by induction over m.
Hereafter, we shall discuss some relevant properties of the new brackets (14).
Consequently, (18) is obtained by induction over m, starting with the case m = 1 already proved in Eq. (9). Similarly, property (19) can be proved by induction over m ≥ 2; the case m = 2 simply requires a direct calculation. Equation (20) is an immediate consequence of Eqs. (18) and (19).
Proposition 13. Let M be a differentiable manifold and A, B : X(M ) → X(M ) two commuting operators. For any f, g, h, k ∈ C ∞ (M ) , X, Y ∈ X(M ) and for each integer m ≥ 2, we have The formula can be proved by induction over m, starting with the case h = 0 and k = 1. Then, the result follows as a consequence of the symmetry w.r.t. the interchange of the first and second operator. hold.
Proof. Eqs. (22) and (23) The value m = 1 has been excluded in Eq. (23), since for this case a separate formula for the Nijenhuis torsion holds: This equation can be easily obtained from Eq. (7), choosing f = g and B = A.
Let us consider in more detail the properties of the Haantjes bracket of level m = 2 of two arbitrary commuting operators. Hereafter the notation A,B (X, Y ) will be used. Proposition 15. Let M be a differentiable manifold, f, g ∈ C ∞ (M ) and let A, B : X(M ) → X(M ) be two (1, 1) tensors. Then, the following identity holds: Formula (25) can be derived by a direct (although cumbersome) calculation. From Definition 9, by means of some algebraic manipulations one can derive another useful result. Proof. We denote by A i i and B j j the non-vanishing components of A and B respectively. Then, in a local chart where the operators diagonalize simultaneously, using Eq. (15) we get, by means of a direct calculation, where A, B i jk is explicitly given in formula (6).
The latter property, which does not hold in the case of the Frölicher-Nijenhuis bracket, is analogous to the one valid for the standard Haantjes torsion of diagonalizable operators. In fact, the Haantjes torsion vanishes, whereas the Nijenhuis one does not necessarily.

Haantjes brackets and Haantjes modules.
In the following analysis, we shall illustrate the algebraic meaning of Haantjes brackets of level 2. As we will show, they play a crucial role in the study of the C ∞ (M )-modules of Haantjes operators, that we shall call Haantjes modules. • M is a differentiable manifold of dimension n; • H M is a set of Haantjes operators K : Thus, a Haantjes module is a free module of Haantjes operators over the ring of smooth functions on M . If property (26) is satisfied only when f, g are real constants, we shall use the denomination of Haantjes vector space.
We determine now the tensorial compatibility conditions ensuring the existence of the Haantjes module generated by two arbitrary Haantjes operators A, B : X(M ) → X(M ). First, we construct these conditions in full generality, namely for non-semisimple, non-commuting Haantjes operators. Then, we shall restrict to the important case of semisimple, commuting operators, which arises for instance in Hamiltonian classical mechanics, in the discussion of separable systems [21], [22].
3.3.1. The general case. We shall start our analysis with the following identity, valid for all f, g ∈ C ∞ (M ), X, Y ∈ X(M ): and From Eqs. (34) we get the identity Some examples of applications are in order.
• Haantjes moduli: In [22], a Haantjes module of operators for the Post-Winternitz superintegrable system has been constructed. The two generators of the module, which do not commute, fulfill conditions (36) and (37). • Haantjes vector spaces: In [8], (2+1)-dimensional hydrodynamic type systems of the form u t = A(u)u x + B(u)u y , where u = u(x, y, t) have been considered. In the case of the generalized Benney system and of an isoentropic gas, the two associated operators A(u) and B(u) do not commute. Also, they fulfil Eq. (36) but not (37). Therefore, these operators generate a Haantjes vector space.  H 1 (A, B) and H 2 (A, B) over two common eigenvectors X µ and Y ν of two (arbitrary) operators A and B (the details of the calculation are reported in Appendix 6.3).
Proposition 22. Let A and B two Haantjes operators and X µ , Y ν two common eigenvectors. Then In addition, if A and B also commute, then Proof. From Eq. (84) in Appendix 6.3 and the assumption that A and B are Haantjes operators it follows that Consequently, it is evident from Eqs. (87)  We can now formulate our main result concerning the characterization of Haantjes modules. the i-th generalized eigen-distribution of index ρ i , that is the distribution of all the generalized eigenvectors corresponding to the eigenvalue λ i . In Eq. (42), ρ i stands for the Riesz index of λ i , which is the minimum integer such that we also assume that ρ i is (locally) independent of x. When ρ i = 1, D i is a proper eigen-distribution. Hereafter, unless differently stated, we shall use the adjective "generalized" to include the case of proper eigen-distributions as well.
In several applications, it is also useful to consider the action of our generalized torsions of any level on the generalized eigenvectors of A. Inspired by a formula for the Nijenhuis torsion evaluated on eigenvectors (proved in Appendix 6.2), we construct a generalized expansion, in terms of commutators, for the torsions of any level. It can be proved by induction over the integers m ≥ 2 via a direct procedure. A be a (1,1) tensor and X α , Y β be two of its generalized eigenvectors of D µ , D ν , respectively. Then, for any integer m ≥ 2 the following formula holds:

Proposition 24. Let
This proposition will be useful in the proof of Lemmas 36 and 38, stated below.

Generalized Nijenhuis torsions and Haantjes brackets for nilcyclic operators
In order to clarify the geometric relevance of both the generalized Nijenhuis torsions and the Haantjes bracket of level m, we shall focus first on the case of nilcyclic operators (namely operators which are both nilpotent and cyclic). According to the classical Jordan-Chevalley decomposition theorem, given a vector space V , any linear endomorphism L : V → V with real eigenvalues can be written in a unique way as the sum L = D + N , where D is a diagonalizable operator and N is a nilpotent operator, commuting with D.
Hereafter, the symbol will denote a C ∞ (M )-linear span of vector fields.

Triangular form of nilcyclic operators.
Let M be an n-dimensional differentiable manifold, and A : X(M ) → X(M ) be a nilcyclic [5] operator, that is a nilpotent (1,1) tensor of maximal index n: This condition implies that there exist local reference frames, possibly non integrable ones, in which A is represented by a single, upper strictly triangular Jordan block. Under these assumptions, the characteristic null flag of A is a complete flag [14], that is, rank(ker A j ) = j , j = 1, . . . , n. Also, the following inclusions hold: Let us assume that there exists a local coordinate chart (x 1 , . . . , x n ) on M where A takes the upper strictly triangular form Here a i j (x) = a i j (x 1 , . . . , x n ) are smooth arbitrary functions depending on the local coordinates on M . In this case, the integrable distributions of the natural flag coincide with the kernels of the powers of the operator A. Precisely, C j = ker A j |U , j = 1, . . . , n .
The following result establishes a necessary condition for a nilcyclic operator to be represented in the upper triangular form, in a suitable coordinate chart.
Proposition 26. Let M be an n-dimensional differentiable manifold, and A : X(M ) → X(M ) be a nilcyclic (1,1) tensor on M . If there exists a local chart where the operator A takes the triangular form (46), then the generalized Nijenhuis torsion of level (k − 1) vanishes for all X, Y ∈ ker A k : Proof. First, we observe that the (strong) invariance conditions hold as a consequence of relations (45) and (47). In the latter conditions, it is understood that C j−p ≡ C 0 for j ≤ p. Then, we can proceed by induction over k = 2, . . . , n − 1. To this aim, notice that for k = 2, we have The first addend vanishes because both e 1 , e 2 are constant fields, the second and third one vanish since e 1 ∈ ker A, whereas the last term is zero due to both the invariance condition (49) and the obvious involutivity of ker A (being rank (ker A) = 1). Now we assume that This hypothesis, jointly with Definition 16 and the A-invariance of ker A k implies τ (k) A (e i , e j ) = 0 , i, j = 1, . . . , k .
We are left with the terms τ (k) A (e i , e k+1 ) , i = 1, . . . , k, which can be evaluated by means of Eq. (17). We obtain As e i ∈ ker A k , the addends corresponding to p = 0 vanish. Moreover, for p > 0, by virtue of equation (49), the following inclusions hold: (51) We can now infer a direct, but important consequence of Proposition (26).
Corollary 27. Let M be an n-dimensional differentiable manifold, n ≥ 2 and A : X(M ) → X(M ) be a nilcyclic (1,1) tensor on M . Then, the condition is necessary for the existence of a local chart where A takes the triangular form (46).
Proof. It is sufficient to apply Proposition (26) to the torsion of level k = n and to observe that ker A n = X(M ), as A is nilcyclic.
Consider the slightly more general case of a tensor of the form where A is a nilcyclic operator. We have the following result.
and  In the first part of the discussion, the eigenvalues and eigenvectors of operators are supposed to be known. However, this hypothesis will be removed in the statement of our Main Theorem: indeed, no knowledge a priori of the spectrum and the eigen-distributions of the operators involved will be assumed.
Remark 30. All the eigen-distributions considered are supposed to be regular, that is they have constant rank on M . For involutive distributions, this condition is equivalent to their Frobenius integrability.
Definition 31. Let us consider a set of distributions {D i , D j , . . . , D k }. We shall say that such distributions are mutually integrable if (i) each of them is integrable; (ii) any sum D i + D j + · · · + D k is also integrable.
First we state the following Lemma 32. Let A : X(M ) → X(M ) be a non-invertible operator. For any X, Y ∈ ker A, we have Proof. Equation (58) comes from Eq. (17) taking into account that the terms with p < m and q < m vanish.
Let us recall that the Riesz index of an operator A is the Riesz index ρ of its zero eigenvalue (supposed to be locally constant over M ), namely the minimum integer ρ that makes stationary the sequence Proposition 33. Let A : X(M ) → X(M ) be an operator and ρ its Riesz index.
The following conditions are equivalent: 1) the distribution ker A ρ is involutive; 2) = ker A ρ if and only if the l.h.s. of Eq. (62) vanishes for some m ∈ N\{0}.
1) =⇒ 3) It is a direct consequence of Eq. (62). The converse statement can be proved by following the same reasoning used in the proof of the first equivalence.
The equivalence of the conditions (2) and (3) can be geometrically interpreted by observing that, if the distribution D = ker A ρ is integrable, then the operator A ρ can be restricted to each integral leaf of D; besides, each of these restricted operators vanishes.
Thus, applying Proposition 33 to each operator B i := A − λ i I, we obtain a novel necessary and sufficient condition for the integrability of the generalized eigen-distributions of an operator with real eigenvalues.
The original Nijenhuis theorem [19] was not stated in the general case of non-semisimple operators. However, the previous analysis allows us to conclude that both the Nijenhuis torsion and the higher-level ones are equally valid, from a theoretical point of view, to detect the integrability properties of the generalized eigen-distributions of a non-semisimple operator.

Main
Theorem. The results stated above provide new necessary and sufficient conditions for the integrability of eigen-distributions of generalized eigenvectors. However, as we have remarked, they require the knowledge a priori of the eigenvalues and eigenvectors of the considered operator. Instead, in the spirit of the seminal theorems by Nijenhuis and Haantjes, it is desirable to have integrability conditions which do not require to solve explicitly eigenvalue problems, since this task is computationally intractable for large values of n. To this aim, we shall propose a novel strategy, based on the notion of generalized Nijenhuis tensors.
Formally, the problem we shall address is the following: to establish the conditions ensuring a priori the integrability of the generalized eigen-distributions of an operator A whose Haantjes torsion does not vanish, without recurring to the explicit determination of its eigen-distributions. To the best of our knowledge, no result is known regarding this problem. In the main Theorem stated below, we will offer a solution to this problem proposing a family of sufficient conditions for integrability.
First, let us prove some preliminary results.
Lemma 36. Let A : X(M ) → X(M ) be an operator, µ ∈ Spec(A) and X α , Y β ∈ D µ two of its generalized eigenvectors, of index α, β respectively, belonging to (possibly different) Jordan chains. If there exists an integer m ≥ 1 such that where min(· , · ) stands for the minimum of its arguments.
Proof. First, we prove the case m ≥ 2. If α = β = 1 and µ = ν, Eq. (44) implies . By induction over (α + β), and applying the operator A − µI α+β−m to both members of Eq. (44) it follows that In order to prove the case m = 1, we observe that if the Nijenhuis torsion τ Proof. Assuming condition (66), Lemma 36 immediately implies that D µ is an involutive distribution, since In the specific case ρ µ = 1, every µ-eigenvector of A is a proper eigenvector, and from Eq. (44) for m ≥ 2 one infers that We deduce that for ρ µ = 1, condition (66) is also necessary for the involutivity of D µ .
Lemma 38. Let A : X(M ) → X(M ) be an operator and D µ , D ν two eigendistributions satisfying, for some integer m ≥ 1, the condition A (D µ , D ν ) = 0 . Then, the commutator of two generalized eigenvectors of A, with respect to two different eigenvalues µ, ν, satisfies the property Proof. If α = β = 1 and µ = ν, Eq. (44) for m ≥ 2 implies that By induction over (α + β), the result follows for m ≥ 2 applying the operator   Proposition 39. Let A : X(M ) → X(M ) be an operator and D µ , D ν two eigendistributions with Riesz indices ρ µ , ρ ν respectively. Assume that for some m ≥ 1, Now, we can prove our main result concerning the mutual integrability of the eigen-distributions of operators.
Theorem 40. Let A : X(M ) → X(M ) be an operator. Assume that for some m ≥ 1. Then, each generalized eigen-distribution of A as well as each direct sum of its eigen-distributions is integrable.
Proof. This result is a direct consequence of Lemma 36 and Proposition 37, whose hypotheses are indeed fulfilled once we assume the validity of condition (71).

Block-diagonalization.
As a nontrivial application of Theorem (40), we shall prove that, given an operator A, condition (71) is also sufficient to ensure the existence of a local chart where the operator A can be block-diagonalized. Potentially relevant applications can be found, for instance, in the theory of hydrodynamictype systems [3], in the study of partial separability of Hamiltonian systems [6] and, more generally, in the context of Courant's problems for first-order hyperbolic systems of partial differential equations [7]. Let A be an operator satisfying condition (71); we denote by r i the rank of the distribution D i of A. We also introduce the distribution (of corank r i ) which is spanned by all the generalized eigenvectors of A, except those associated with the eigenvalue λ i (we remind that A by default has real eigenvalues). We shall say that E i is a characteristic distribution of A. Let E • i denote the annihilator of the distribution E i . The cotangent spaces of M can be decomposed as  (74) Here c i,k are real constants depending on the indices i and k only. In this case, we shall say that the collection of these functions is adapted to the web and that each of them is a characteristic function.
Proposition 43. Let A : X(M ) → X(M ) be an operator. If for some m ≥ 1, then A admits local charts where it takes a block-diagonal form.
Proof. Theorem 40 ensures that each characteristic distribution E i is integrable. Thus, we can also deduce the existence of r i exact one-forms (dx i,1 , . . . , dx i,ri ) in the corresponding annihilator E • i ; consequently, there exist functions x i = (x i,1 , . . . , x i,ri ) adapted to the characteristic foliation E i . Collecting together all these functions, we get n coordinates (x 1 , . . . , x i , . . . , x s ) and therefore, a local chart {U, (x 1 , . . . , x i , . . . , x s )}, adapted to the characteristic web. The natural frame associated ∂ ∂x 1 , . . . , ∂ ∂x i , . . . , ∂ ∂x s is a generalized eigen-frame. To prove this, it is sufficient to observe that the following decomposition holds: Thus, any generalized eigenvector W ∈ D i leaves invariant all the coordinate functions except at most the characteristic functions x i = (x i,1 , . . . , x i,ri ) of E i . Thus, we deduce that This means that each frame equivalent to is an integrable eigen-frame of generalized eigenvectors. Consequently, there exists an equivalence class of integrable frames, with their local charts associated. In these charts, the operator A, due to the invariance of its eigen-distributions, takes a block-diagonal form.

5.4.
A comparison with Haantjes's classical theorem. In his seminal paper [11], Haantjes proved the following, fundamental theorem: i) If A is a semisimple operator, the vanishing of its Haantjes torsion is a necessary and sufficient condition for the integrability of all of its eigendistributions and direct sums of them.
ii) If A is non-semisimple, then condition (78) is sufficient to guarantee the integrability of its generalized eigen-distributions, but it is not necessary.
Our improvement of the Haantjes theorem consists in the family of conditions (71), which indeed are more general than the standard vanishing condition of the Haantjes torsion. Indeed, given a non-semisimple operator A, no conclusion about integrability of its eigen-distributions can be deduced from the Haantjes theorem, if H A (X, Y ) = 0. However, if there exists m > 2 such that τ (m) A (X, Y ) = 0, this weaker condition is sufficient to ensure integrability.
In the semisimple case, ρ i = 1 ∀i = 1, . . . , s, so we recover Haantjes's result on integrability directly from Proposition 37. Instead, in the most general, nonsemisimple case (ρ i > 1), Theorem 40 provides an infinite family of new sufficient conditions.
The following, simple example can illustrate the potential relevance of Theorem 40 in applicative contexts. Indeed, already in the case n = 3 a generic non-semisimple operator is not necessarily a Haantjes one. Therefore, the Haantjes theorem does not apply. However, in our example, the associated generalized tensor of level three vanishes; this ensures integrability.
Example. Let M be a 3 dimensional manifold and (x 1 , x 2 , x 3 ) a local chart in M . Consider the operator with λ 1 , λ 2 , f, g ∈ C ∞ (M ), λ 1 = λ 2 . A direct calculation shows that, for generic choices of these functions, the Nijenhuis and Haantjes torsions do not vanish identically; however, τ L (X, Y ) = 0. Therefore, according to Theorem 40, the generalized eigen-distributions of L are mutually integrable. To construct them explicitly, observe that the minimal polynomial of L is m(λ) := (λ − λ 1 ) 2 (λ − λ 2 ), so that the Riesz indices of λ 1 and λ 2 are ρ 1 = 2, ρ 2 = 1, respectively. We obtain the generalized eigen-distribution D 1 = ker(L − λ i I) 2 = ∂ ∂x 1 , ∂ ∂x 2 , which is trivially integrable, as well as the proper eigen-distribution D 2 = ker(L − λ 2 I) = X λ2 , with X λ2 = f g ∂ ∂x 1 + (λ 2 − λ 1 )g ∂ ∂x 2 + (λ 1 − λ 2 ) 2 ∂ ∂x 3 . The latter eigen-distribution is of rank 1 and obviously integrable. Thus, as D 1 = E 2 and D 2 = E 1 , we get the spectral decompositions of the tangent spaces T x M = D 1 ⊕ E 1 = D 2 ⊕E 2 . Correspondingly, for the cotangent spaces, we obtain T * x M = E • 1 ⊕E • 2 , where the annihilators of the characteristic distributions of L are In order to construct explicitly a local chart where L takes a block-diagonal form (as ensured by Proposition 43), let us consider the space R 3 endowed with Cartesian coordinates (x 1 , x 2 , x 3 ). We make the simple choice λ 1 = x 1 +x 2 +x 3 , λ 2 = x 1 +x 2 , f = x 3 , g = x 1 in M = R 3 \{x 3 = 0} (to guarantee λ 1 = λ 2 ). By integrating the annihilators of the characteristic distributions (as explained in the proof of Proposition 43), we find the local coordinate chart On this chart, the operator L takes the block-diagonal form As we have shown, in the case of non-semisimple operators, the criterion of the vanishing of the Haantjes torsion, being only sufficient, may fail to detect the mutual integrability of the eigen-distributions even for very basic examples. Nevertheless, Theorem 40 provides us with a more general tensorial test, guaranteeing integrability without the need for an explicit analysis of the eigen-distributions involved. Once integrability is ascertained, one can enter this kind of analysis in order to block-diagonalize the considered operator. 6.2. The Nijenhuis torsion evaluated over eigenvectors. Let A be an operator. Without loss of generality, we shall focus only on two eigenvalues of A, µ = µ(x) and ν = ν(x) ∈ Spec(A), possibly coincident. Let us denote by X α , Y β two generalized eigenvectors, with indices α and β, associated with µ and ν: (81) . They belong to certain Jordan chains defined in D µ , D ν , respectively: (82) AX α = µX α + X α−1 , where X 0 and Y 0 are, by definition, null vector fields. Evaluating the Nijenhuis torsion on such eigenvectors, we obtain − X α (ν)Y β−1 + Y β−1 (µ)X α + X α−1 (ν)Y β + Y β (µ)X α−1 .
The Frolicher-Nijenhuis bracket satisfies the identity Thus, we get