  The paper deals with singular first order Hamiltonian systems of the form \[
\Gamma_k\dot{z}_k(t)=J\nabla_{z_k} H\big(z(t)\big),\quad z_k(t) \in \Omega
\subset \mathbb{R}^2,\ k=1,\dots,N, \] where $J\in\mathbb{R}^{2\times2}$
defines the standard symplectic structure in $\mathbb{R}^2$, and the
Hamiltonian $H$ is of $N$-vortex type: \[ H(z_1,\dots,z_N) = -\frac1{2\pi}
\sum_{j\neq k=1}^N \Gamma_j \Gamma_k \log|z_j-z_k| - F(z). \] This is defined
on the configuration space $\{(z_1,\ldots,z_N)\in \Omega^{2N}:z_j\neq z_k\text{
for }j\neq k\}$ of $N$ different points in the domain
$\Omega\subset\mathbb{R}^2$. The function $F:\Omega^N\to\mathbb{R}$ may have
additional singularities near the boundary of $\Omega^N$. We prove the
existence of a global continuum of periodic solutions
$z(t)=(z_1(t),\dots,z_N(t))\in\Omega^N$ that emanates, after introducing a
suitable singular limit scaling, from a relative equilibrium
$Z(t)\in\mathbb{R}^{2N}$ of the $N$-vortex problem in the whole plane (where
$F=0$). Examples for $Z$ include Thomson's vortex configurations, or
equilateral triangle solutions. The domain $\Omega$ need not be simply
connected. A special feature is that the associated action integral is not
defined on an open subset of the space of $2\pi$-periodic $H^{1/2}$ functions,
the natural form domain for first order Hamiltonian systems. This is a
consequence of the singular character of the Hamiltonian. Our main tool in the
proof is a degree for $S^1$-equivariant gradient maps that we adapt to this
class of potential operators.
