  It is known that knowledge of a symmetry of a scalar Ito stochastic
differential equations leads, thanks to the Kozlov substitution, to its
integration. In the present paper we provide a classification of scalar
autonomous Ito stochastic differential equations with simple noise possessing
symmetries; here "simple noise" means the noise coefficient is of the form $\s
(x,t) = s x^k$, with $s$ and $k$ real constants. Such equations can be taken to
a standard form via a well known transformation; for such standard forms we
also provide the integration of the symmetric equations. Our work extends
previous classifications in that it also consider recently introduced types of
symmetries, in particular standard random symmetries, not considered in those.
