  For the generalized surface quasi-geostrophic equation $$\left\{
\begin{aligned} & \partial_t \theta+u\cdot \nabla \theta=0, \quad \text{in }
\mathbb{R}^2 \times (0,T), \\ & u=\nabla^\perp \psi, \quad \psi =
(-\Delta)^{-s}\theta \quad \text{in } \mathbb{R}^2 \times (0,T) , \end{aligned}
\right. $$ $0<s<1$, we consider for $k\ge1$ the problem of finding a family of
$k$-vortex solutions $\theta_\varepsilon(x,t)$ such that as $\varepsilon\to 0$
$$ \theta_\varepsilon(x,t) \rightharpoonup \sum_{j=1}^k m_j\delta(x-\xi_j(t))
$$ for suitable trajectories for the vortices $x=\xi_j(t)$. We find such
solutions in the special cases of vortices travelling with constant speed along
one axis or rotating with same speed around the origin. In those cases the
problem is reduced to a fractional elliptic equation which is treated with
singular perturbation methods. A key element in our construction is a proof of
the non-degeneracy of the radial ground state for the so-called fractional
plasma problem $$(-\Delta)^sW = (W-1)^\gamma_+, \quad \text{in } \mathbb{R}^2,
\quad 1<\gamma < \frac{1+s}{1-s}$$ whose existence and uniqueness have recently
been proven in \cite{chan_uniqueness_2020}.
