TITLE: "Noncommutative heat equation" -- a strange generalization of Killing vectors for a flat metric
QUESTION [8 upvotes]: Let $(M,g)$ be a smooth (pseudo)Riemannian manifold with a flat metric $g$, and $X$, $Y$ be vector fields on $M$ such that
$$
L_X^2 (g)=L_Y(g). \hspace{70mm} \mbox{(1)}
$$ 
where $L_Z$ is the Lie derivative along $Z$ and $L_X^2(g)\equiv L_X(L_X(g))$.
If $X=0$ then $Y$ is just a Killing vector for $g$ but 

is there any (geometric?) interpretation for $X$ and $Y$ in the general case?

In fact I wonder whether Eq.(1), be it for fixed $g$ and unknown $X,Y$ or conversely for fixed $X,Y$ and unknown $g$, was studied systematically at all: I failed to find any relevant references.
Motivation: Equation (1) follows from the last formula in Proposition 5 from a paper on classification of compatible Hamiltonian structures that I have recently come across. It looks a bit like the heat equation for the metric, hence the title.

REPLY [8 votes]: To understand the local geometry of this equation, I think one should apply the Calabi resolution of the Killing equation.  (See E. Calabi, On compact, Riemannian manifolds with constant curvature. I, in Differential Geometry, C. B. Allendoerfer, ed., vol. 3 of Proceedings of Symposia in Pure Mathematics, pp. 155–180. AMS, Providence, RI, 1961.  For a useful modern exposition, see http://arxiv.org/pdf/1409.7212.pdf.)
The basic point is that, when $g$ is flat (or, more generally, has constant sectional curvature) the Killing operator $X\mapsto D_1(X) =  L_X(g)$ as a map from $C^\infty(TM)$ to $C^\infty(S^2(T^*M))$ can be embedded into a locally exact sequence of sheaves
$$
0 \longrightarrow \mathscr{K}_g\longrightarrow C^\infty(TM)\longrightarrow C^\infty(S^2(T^*M))\longrightarrow C^\infty\bigl(C^{(2,2)}(TM)\bigr)\longrightarrow\cdots\tag1
$$
that resolves the sheaf $\mathscr{K}_g$ of $g$-Killing vector fields on $M$ and where the sheaf mappings are given by known linear differential operators.  An element $h$ of $C^\infty(S^2(T^*M))$ can be written (locally) in the form $h = L_Y(g)$ for some vector field $Y$ if and only if $D_2(h) = 0$.  
This shows that if $L_X(L_X(g)) = L_Y(g)$, then $X$ must satisfy the system of differential equations
$$
D_2\bigl(L_X(L_X(g))\bigr) = 0.\tag2
$$
Conversely, if a vector field $X$ satisfies this equation, then, at least locally (and globally if $M$ is simply-connected), there will exist a vector field $Y$ such that $L_X(L_X(g)) = L_Y(g)$, and this $Y$ will be unique up to the addition of a Killing vector field.
The differential equation (2) for $X$ is generally overdetermined (and nonlinear) when the dimension $n$ of $M$ is greater than $2$, and one expects the space of its local solutions to be finite dimensional.  Thus, one expects that, even locally, the space of solutions of the original system for $(X,Y)$ will be finite dimensional when $n>2$.  To check this, one should compute the characteristic variety of (2).  If there is interest, I can do this and report the result.