In this paper we study the Stratonovich stochastic differential equation $\mathrm{d} X=|X|^��\circ\mathrm{d} B$, $��\in(-1,1)$, which has been introduced by Cherstvy et al. [New Journal of Physics 15:083039 (2013)] in the context of analysis of anomalous diffusions in heterogeneous media. We determine its weak and strong solutions, which are homogeneous strong Markov processes \chng{spending zero time at $0$: for $��\in (0,1)$, these solutions have the form $$ X_t^��=\bigl((1-��)B_t^��\bigr)^{1/(1-��)}, $$ where $B^��$ is the $��$-skew Brownian motion driven by $B$ and starting at $\frac{1}{1-��}(X_0)^{1-��}$, $��\in [-1,1]$,} and $(x)^��=|x|^��\operatorname{sign} x$; for $��\in(-1,0]$, only the case $��=0$ is possible. The central part of the paper consists in the proof of the existence of a quadratic covariation $[f(B^��),B]$ for a locally square integrable function $f$ and is based on the time-reversion technique for Markovian diffusions.