Automorphism groups of rigid affine surfaces: the identity component

It is known that the identity component of the automorphism group of a projective algebraic variety is an algebraic group. This is not true in general for quasi-projective varieties. In this note we address the question: given an affine algebraic surface $Y$, as to when the identity component ${\mathrm{A}\mathrm{u}\mathrm{t}}^0(Y)$ of the automorphism group $\mathrm{A}\mathrm{u}\mathrm{t}(Y)$ is an algebraic group? We show that this happens if and only if $Y$ admits no effective action of the additive group of the field. In the latter case, ${\mathrm{A}\mathrm{u}\mathrm{t}}^0(Y)$ is an algebraic torus of rank $\le 2$.