  In general, adding a stochastic perturbation to a differential equation
possessing an invariant manifold destroys the invariance as far as the It\^o
formalism is used. In this article, we propose an invariantization method for
perturbations in the It\^o case which can be used to restore invariance. We
then apply our results to develop a stochastic version of the Landau-Lifshitz
equation. We discuss in particular previous results obtained by Etore and al.
in [P. \'Etor\'e, S.Labb\'e , J. Lelong, Long time behaviour of a stochastic
nanoparticle, J. Differential Equations 257 (2014), 2115-2135].
