Suppose I have some tensor equation which deals with rank-2 tensors of two variables $x,y$,
$$ \frac{\partial K^{ab}}{\partial x^k} g^{kc} = \frac{\partial g^{ab}}{\partial x^k} K^{kc} $$
This is obviously satisfied for $K^{ab} = g^{ab}$, but is there a way to solve the equation generally for $K^{ab}$?