  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 ${\rm Aut}^0
(Y)$ of the automorphism group ${\rm Aut} (Y)$ is an algebraic group? We show
that this happens if and only if $Y$ admits no effective action of the additive
group. In the latter case, ${\rm Aut}^0 (Y)$ is an algebraic torus of rank $\le
2$.
