On the algebraic models of symmetric smooth manifolds

Let M be a compact smooth submanifold of R^n symmetric with respect to an affine subspace L. In this paper, we prove that M has a symmetric Nash model N, which is unique up to Nash isomorphism preserving symmetries. Such a model N can be chosen equal to a union of nonsingular connected components of a real algebraic set if M is a (not necessarily compact) Nash submanifold of R^n. If M does not intersect the mirror L, then we are able to construct a symmetric algebraic model of M too. More precisely, in this case, we show that the set of birationally nonisomorphic symmetric algebraic models of M has the power of continuum