On Solvable R* Groups Of Finite Rank

For an element a of a group G, let S(a) denote the semigroup generated by all conjugates of a in G. We prove that if G is solvable of finite rank and 1 is an element of S(a) for all 1 not equal a is an element of G, then / is a periodic group for every b is an element of S(a). Conversely if every two generator subgroup of a finitely generated torsion-free solvable group G has this property then G has finite rank, and if every finitely generated subgroup has this property then every partial order on G can be extended to a total order.