On the harmonic M\"obius transformations

It is well-known that two locally univalent analytic functions $\phi$ and $\psi$ have equal Schwarzian derivative if and only if there exists a non-constant M"obius transformation $T$ such that $\phi =T\circ \psi$. In this paper, we identify completely the relationship between two locally univalent harmonic mappings with equal (harmonic) Schwarzian derivative.