Consider the pair $(g, L)$, where $g$ is a pseudo-Riemannian (i.e., nondegenerate and of arbitrary signature) metric and $L$ is an (1,1) $g$-selfadjoint tensor field.

Does there exist a local description of such pairs; for example in the form "in a certain coordinate system $g$ and $L$ are given by formulas ...".

For me, the state of art is as follows: for Riemannian metrics, the answer was known to classics: The existence of such $L$ implies the local decomposition of $g$ into a direct product: $g= g_0+ ... + g_k$, where $g_0$ is flat (we can think therefore that $g_0= dx_1^2 +...+dx_m^2$ in a certain coordinate system) and each $g_i$ has irreducible holonomy group. For such metrics $g= g_0+ ... + g_k$, the tensor $L$ also can be decomposed into the product $L= L_0+...+L_k$; moreover, $L_0$ is given by arbitrary symmetric matrix with constant entries and other $L_i$ are proportional to identity with constant coefficients (the coefficients depend on the component).

For the pseudo-Riemannian metrics one can do the same splitting if $L$ has different eigenvalues; so the interesting case is when $L$ has one real eigenvalue or two conjugated complex eigenvalues. And this case looks quite open for me; only the special case when $L^2= 0$ or $L^2= -1$ are known.

Does anybody know more?