Affine Extensions of Principal Additive Bundles over a Punctured Surface

The aim of this article is to make a first step towards the classification of complex normal affine $G_a$-threefolds $X$. We consider the case where the restriction of the quotient morphism $picolon Xto S$ to $pi^{-1}(S_*)$, where $S_*$ denotes the complement of some regular closed point in $S$, is a principal $G_a$-bundle. The variety $SL_2$ will be of special interest and a source of many examples. It has a natural right $G_a$-action such that the quotient morphism $SL_2toA^2$ restricts to a principal $G_a$-bundle over the punctured plane $A^2_*$.