On triviality of ${\mathbb{A}}^2$-forms admitting a nontrivial ${\mathbb{G}}_a$-action

T. Kambayashi had shown that ${\mathbb{A}}^2$-forms over separable field extensions are necessarily polynomial rings. However, there exist inseparable ${\mathbb{A}}^2$-forms which are not necessarily polynomial rings. In this paper, we give a structure theorem for ${\mathbb{A}}^2$-forms over arbitrary field extensions admitting a nontrivial ${\mathbb{G}}_a$-action. From this structure theorem we derive some conditions under which an ${\mathbb{A}}^2$-form becomes trivial. In particular, we prove that over a field $k$, a factorial ${\mathbb{A}}^2$-form having a $k$-rational point and a non-trivial ${\mathbb{G}}_a$-action is trivial and we also give examples demonstrating that none of these hypotheses can be discarded. As a consequence of the structure theorem, we obtain a generalization of the Zariski Cancellation Theorem for the affine plane over an arbitrary field.