Additive group actions on affine TT-varieties of complexity one in arbitrary characteristic

Let X   be a normal affine TT-variety of complexity at most one over a perfect field k, where T=Gmn stands for the split algebraic torus. Our main result is a classification of additive group actions on X   that are normalized by the TT-action. This generalizes the classification given by the second author in the particular case where k is algebraically closed and of characteristic zero.With the assumption that the characteristic of k is positive, we introduce the notion of rationally homogeneous locally finite iterative higher derivations which corresponds geometrically to additive group actions on affine TT-varieties normalized up to a Frobenius map. As a preliminary result, we provide a complete description of these GaGa-actions in the toric situation.