  In this paper, we study the radial symmetry properties of stationary and
uniformly rotating solutions of the vortex-wave system introduced by Marchioro
and Pulvirenti \cite{Mar1}. We show that every uniformly rotating patch
$\left(D,x_1,x_2,..,x_k\right)$ with angular velocity $\Omega\leq 0$ must be
radial with respect to the only point vortex $x_1$, implying that $k=1$. In
other words, the background vorticity consists of finite nested annulus and the
point vortex is located at the center of these annulus. In contrast to the case
where the angular velocity is non-positive, we prove that there exists a family
of uniformly rotating patch $(D^n, x_1^n)_n$ solutions, which are associated
with a sequence of positive angular velocities $\{\Omega_n\}$ and are not
annular. Furthermore, we find that the set of bifurcating angular velocities
$\{\Omega_n\}$ is dense in the interval $(0,+\infty)$, a novel feature that
distinguishes this behavior from that observed in the classical Euler equation
and gSQG equation.
