  This paper is devoted to the study of qualitative geometrical properties of
stochastic dynamical systems, namely their symmetries, reduction and
integrability. In particular, we show that an SDS which is diffusion-wise
symmetric with respect to a proper Lie group action can be diffusion-wise
reduced to an SDS on the quotient space. We also show necessary and sufficient
conditions for an SDS to be projectable via a surjective map. We then introduce
the notion of integrability of SDS's, and extend the results about the
existence and structure-preserving property of Liouville torus actions from the
classical case to the case of integrable SDS's. We also show how integrable
SDS's are related to compatible families of integrable Riemannian metrics on
manifolds.
