  An affine algebraic variety X of dimension at least 2 is called flexible if
the subgroup SAut(X) in Aut(X) generated by the one-parameter unipotent
subgroups acts m-transitively on reg(X) for any m $\ge$ 1. In the previous
paper we proved that any nondegenerate toric affine variety X is flexible. In
the present paper we show that one can find a subgroup of SAut(X) generated by
a finite number of one-parameter unipotent subgroups which has the same
transitivity property, provided the toric variety X is smooth in codimension 2.
For X=$\mathbb{A}^n$ with n$\ge$2, three such subgroups suffice.
