  Our main result is the following: let X be a normal affine toric surface
without torus factor. Then there exists a non-normal affine toric surface X'
with automorphism group isomorphic to the automorphism group of X if and only
if X is different from the affine plane. As a tool, we first provide a
classification of normalized additive group actions on a non-necessarily normal
affine toric variety X of any dimension. Recall that normalized additive group
actions on X are in correspondence with homogeneous locally nilpotent
derivations on the algebra of regular functions of X. More generally, we
provide a classification of homogeneous locally nilpotent derivations on the
semigroup algebra of a commutative cancellative monoid.
