I find myself staring blankly at a system of PDEs in $n$ dimensions which has "one equation per component" of the Hessian of the unknown function - that is, it specifies the Hessian in terms of the derivative and the value of the function:

$\qquad \partial_i \partial_j f + B_{ij}^{\ \ \ k} \partial_k f + A_{ij} f = 0$

Here, both $A_{ij}(x)$ and $B_{ij}^{\ \ \ k}(x)$ are symmetric in $i,j$ and there is an implicit $k$ summation - my question is simply this:

- Are there further integrability conditions, or is there generically a solution for $f(x)$ given e.g. its value and gradient at the origin?

Apologies if this is trivial, but I've got a complete mental block.