The set of fixed points of a unipotent group

Let K be an algebraically closed field. Let G be a non-trivial connected unipotent group, which acts effectively on an affine variety X. Then every non-empty component R of the set of fixed points of G is a K-uniruled variety, i.e., there exist an affine cylinder W×K and a dominant, generically-finite polynomial mapping ϕ:W×K→R. We show also that if an arbitrary infinite algebraic group G acts effectively on Kn and the set of fixed points contains a hypersurface H, then this hypersurface is K-uniruled.