Recovering of a mapping via Jacobi matrix, normalized homogeneous function

Consider system of the differential equations $f'(x)=\Phi(f'(x))M(x)$ with generalized partial derivatives,where $f'(x)$ is a matrix Jacobi of sought mapping, $M$ is a given $n\times n$ matrix-value function with integrable elements, $\Phi$ is a given function of matrices.