We discuss the interrelations between symmetry of an Ito stochastic
differential equations (or systems thereof) and its integrability, extending in
party results by R. Kozlov [J. Phys. A 43 (2010) \& 44 (2011)].
Together with integrability, we also consider the relations between symmetries
and reducibility of a system of SDEs to a lower dimensional one. We consider
both "deterministic" symmetries and "random" ones, in the sense introduced
recently by Gaeta and Spadaro [J.Math. Phys. 58 (2017)].Comment: to appear in J. Nonlin. Math.Phys. (2018