Cyclic covers of affine TT-varieties

We consider normal affine TT-varieties X endowed with an action of finite abelian group G   commuting with the action of TT. 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 A3A3 equipped with a hyperbolic GmGm-action.