Normal equivariant compactifications of G²_a with Picard number one

We classify all normal G^2_a-surfaces with Picard number one, and characterize which of these surfaces have at worst log canonical, and which have at worst log terminal singularities, answering a question of Hassett and Tschinkel (Int. Math. Res. Not., 1999). We also find all G^2_a-structures on these surfaces and show that these surfaces and their minimal desingularizations have the same G^2_a-structures (modulo equivalence of G^2_a-actions). In particular, we show that some of these surfaces admit one dimensional moduli of G^2_a-structures, answering another question of Hassett and Tschinkel (Int. Math. Res. Not., 1999).