Sufficient conditions for local quasiconformality of mappings with bounded distortion

Sufficient conditions for a mapping $f(x)$ with bounded distortion to be a local homeomorphism are established in terms of estimates of the oscillation of its normalized Jacobi matrix
$$ f'(x)/\lvert\det f'(x)\rvert^{1/n}. $$
The results obtained are used in the description of properties of the solutions of the system of partial differential equations
$$ f'(x)=K(x)\lvert\det f'(x)\rvert^{1/n}, $$
where $K(x)$ is a given matrix-valued function.