We give an algebraic characterization of a log affine pseudo-plane and the quotient surface $\mathbb A^2/G$ of $\mathbb A^2$ with a small finite subgroup $G$ of $GL(2,\mathbb C)$ in terms of $\mathbb Q$-factorial normal subalgebras of a polynomial ring $\mathbb C[x, y]$. Then we consider the cancellation problem for these surfaces.