  The article demonstrates the procedure how to compute the Zariski closure of
an orbit by an algebraic action of finitely generated group on the affine
plane. First half of the algorithm is about deciding whether given finitely
generated group is contained in an algebraic group. For the next half, we
compute the totality of the invariant subvarieties for a single triangular
automorphism. Then, the computation for the individual generators is applied to
compute the orbit Zariski closure for a finitely generated group.
