Motivated by the study of the structure of algebraic actions of the additive group on affine threefolds X, we consider a special class of such varieties whose algebraic quotient morphisms X -> X//G(a) restrict to principal homogeneous bundles over the complement of a smooth point of the quotient. We establish basic general properties of these varieties and construct families of examples illustrating their rich geometry. In particular, we give a complete classification of a natural subclass consisting of threefolds X endowed with proper G(a)-actions, whose algebraic quotient morphisms pi : X -> X//G(a) are surjective with only isolated degenerate fibers, all isomorphic to the affine plane A(2) when equipped with their reduced structures.