  We study the action of the groups $H(\lambda)$ generated by the linear
fractional transformations $x:z\mapsto -\frac{1}{z} \text{ and }w:z\mapsto
z+\lambda$, where $\lambda$ is a positive integer, on the subsets $\mathbb
Q^*(\sqrt{n})=\{\frac{a+\sqrt n}{c}\;|\;a,b=\frac{a^2-n}{c},c\in\mathbb Z\}$,
where $n$ is a square-free integer. We prove that this action has a finite
number of orbits if and only if $\lambda=1$ or $\lambda=2$, and we give an
upper bound for the number of orbits for $\lambda=2$.
