  We consider the Calogero-Sutherland derivative nonlinear Schr\"odinger
equation \begin{equation}\tag{CS}
  i\partial_tu+\partial_x^2u\,\pm\,\frac{2}{i}\,\partial_x\Pi(|u|^2)u=0\,,\qquad
x\in\mathbb{T}\,, \end{equation} where $\Pi$ is the Szeg\H{o} projector
$$\Pi\Big(\sum_{n\in \mathbb{Z}}\widehat{u}(n)\mathrm{e}^{inx}\Big)=\sum_{n\geq
0 }\widehat{u}(n)\mathrm{e}^{inx}\,.$$ First, we characterize the traveling
wave $u_0(x-ct)$ solutions to the defocusing equation (CS$^-$), and prove for
the focusing equation (CS$^+$), that all the traveling waves must be either the
constant functions or plane waves or rational functions. A noteworthy
observation is that the (CS)-equation is one of the fewest nonlinear PDE
enjoying nontrivial traveling waves with arbitrary small and large $L^2$-norms.
Second, we study the finite gap potentials, and show that they are also
rational functions, containing the traveling waves, and they can be grouped
into sets that remain invariant under the system's evolution.
