TITLE: twisted Poisson structures, degenerate metrics and integrability properties of (2,0)-tensors
QUESTION [9 upvotes]: Given a regular (constant rank) bi-vector $\Pi \in \Gamma(\bigwedge^2TM)$ on a smooth manifold $M$ the necessary and sufficient condition for the image of $\Pi^\sharp:T^*M\to TM$ to be an integrable distribution is that $\Pi$ is a twisted Poisson tensor with respect to a closed 3-form $\phi$, i.e. there exist a closed 3-form $\phi$ such that
$$[\Pi,\Pi]=\Pi^\sharp(\phi)$$
(see e.g. arXiv:1104.0880).
What is the corresponding condition for a degenerate regular symmetric (2,0)-tensor (a degenerate metric) $g\in \Gamma(S^2TM)$ to produce an integrable distribution as the image of $g^\sharp:T^*M\to TM$?
And what about any regular (constant rank) tensor $T\in \Gamma(TM\otimes TM)$, with no special symmetry properties?

REPLY [2 votes]: This is also only for the symmetric case.
Let $S(TM)=\bigoplus_k S^k(TM)$ and $\Gamma(S(TM))$ be the commutative graded algebra of symmetric purely contravariant tensor fields. This is the algebra of functions on $T^*M$ which are polynomial on each fiber, and it is invariant under the Poisson bracket on $T^*M$.
This is the symmetric Schouten bracket.
See the following paper for more information: 
Michel Dubois-Violette, Peter W. Michor: A common generalization of the Frölicher-Nijenhuis bracket and the Schouten bracket for symmetric multi vector fields, Indagationes Math. N. S. 6 (1995), 51--66.
(pdf)
It might be that one can just carry over the considerations from $\Gamma(\Lambda (TM))$
to $\Gamma(S(TM))$ using this bracket.
EDIT: The above did not work. So let me try again. I do not need symmetry now.
Suppose that $a\in\Gamma(TM\otimes TM)$ is of constant rank when viewed as 
$a:T^\star M\to TM$.
Choose a Riemann metric $g$ on $M$ and consider $A=a\circ g:TM\to T^\star M\to TM$ which is again of constant rank. So $\ker(A)$ and $A(TM)$ are subbundles. Choose an isomorphism $B:TM\to TM$
such that $A.B$ equals the $g$-orthogonal projection $P$ onto $A(TM)$. $P$ can be canonically constructed from $g$ and $a$, up to some free choice of an isomorphism between $Im(A)^\bot$ and $\ker(A)$. 

Now the Frölicher-Nijenhuis bracket of the vector valued 1-form $P$ is of the form
$$ [P,P] = 2R + 2\bar R,\quad R(X,Y)= P[(Id-P)X,(Id-P)Y], \quad \bar R(X,Y)=(Id-P)[PX,PY]
$$
$R$, the curvature, is exacly the obstruction against integrability of the kernel of $P$.

$\bar R$, the cocurvature, is exactly the obstruction against integrability of the image of $P$, which we are interested in. 

So the condition is $\bar R=0$ which can be written as 
$$
0=P^*[P,P]=[P,P].(P\times P):\Lambda^2TM \to TM.
$$