  We consider normal affine T-varieties X endowed with an action of finite
abelian group G commuting with the action of T. For such varieties we establish
the existence of G-equivariant geometrico-combinatorial presentations in the
sense of Altmann and Hausen. As an application, we determine explicit
presentations of the Koras-Russell threefolds as bi-cyclic covers of A^3
equipped with a hyperbolic C^*-action.
