  For every diffeomorphism $\varphi:M\to N$ between 3--dimensional Riemannian
manifolds $M$ and $N$ there are in general locally two 2--dimensional
distributions $D_{\pm}$ such that $\varphi$ is conformal on both of them. We
state necessary and sufficient conditions for a distribution to be one of
$D_{\pm}$. These are algebraic conditions expressed in terms of the
self-adjoint and positive definite operator $(\varphi_{\ast})^*\varphi_{\ast}$.
We investigate integrability condition of $D_+$ and $D_-$. We also show that it
is possible to choose coordinate systems in which leafwise conformal
diffeomorphism is holomorphic on leaves.
