A note on 2-dimensional Finsler manifold

We solve the following problem for $n=2:$ Is any n-dimensional Finsler manifold $(M, F)$ with a function $f$ which is nonconstant and smooth on $M$ satisfying $\dfrac{\partial g^{ij}}{\partial y^k}\dfrac{\partial f}{\partial x^i}=0,$ a Riemannian manifold? The problem for $n>2$ remains open.