Suppose that f = (u, v) is a homeomorphism in the plane of the Sobolev class W 1,1 loc such that its inverse is of the same Sobolev class. We prove that u and v have the same set of critical points. As an application we show that u and v are distributional solutions to the same non-trivial degenerate elliptic equation in divergence form. We study similar properties also in higher dimensions.