  There are two-dimensional expanding shift radix systems (SRS) which have some
periodic orbits. The aim of the present paper is to describe such unusual
points as well as possible. We give all regions that contain parameters the
corresponding SRS of which generate obvious cycles like $(1), (-1), (1,-1),
(1,0), (-1,0)$. We prove that if $\mathbf{r}=(r_0,r_1)\in \mathbb{R}^2$ neither
belongs to the aforementioned regions nor to the finite region $1\le r_0\le
4/3, -r_0 \le r_1 <r_0-1$, then $\tau_{\mathbf{r}}$ only has the trivial
bounded orbit $\mathbf{0}$, which is a natural generalization of the
established finiteness property for SRS with non-periodic orbits. The further
reduction should be quite involving, because for all $1\le r_0< 4/3$ there
exists at least one interval $I$ such that for the point $(r_0,r_1)$ this is
not true whenever $r_1\in I$.
