Structure theorems for rings under certain coactions of a Hopf algebra

Let $\{D_1, \dots, D_n\}$ be a system of derivations of a $k$-algebra $A$, $k$ a field of characteristic $p > 0$, defined by a coaction $\delta$ of the Hopf algebra $H_c = k[X_1, \dots, X_n]/(X_1^p, \dots, X_n^p)$, $ c \in \{0,1\}$, the Lie Hopf algebra of the additive group and the multiplicative group on $A$, respectively. If there exist $x_1, \dots, x_n \in A$, with the Jacobian matrix $(D_i(x_j))$ invertible, $[D_i,D_j] = 0$, $D_i^p = cD_i$, $c \in \{0, 1\}$, $1 \leq i, j \leq n$, we obtain elements $y_1, \dots, y_n \in A$, such that $D_i(y_j) = \delta_{ij}(1 + cy_i)$, using properties of $H_c$-Galois extensions. A concrete structure theorem for a commutative $k$-algebra $A$, as a free module on the subring $A^{\delta}$ of $A$ consisting of the coinvariant elements with respect to $\delta$, is proved in the additive case.