  In order to circumvent the loss of solid material through radial drift
towards the central star, the trapping of dust inside persistent vortices in
protoplanetary discs has often been suggested as a process that can eventually
lead to planetesimal formation. Although a few special cases have been
discussed, exhaustive studies of possible quasi-steady configurations available
for dust-laden vortices and their stability have yet to be undertaken, thus
their viability or otherwise as locations for the gravitational instability to
take hold and seed planet formation is unclear. In this paper we generalise and
extend the well known Kida solution to obtain a series of steady state
solutions with varying vorticity and dust density distributions in their cores,
in the limit of perfectly coupled dust and gas. We then present a local
stability analysis of these configurations, considering perturbations localised
on streamlines. Typical parametric instabilities found have growthrates of
$~0.05\Omega_P$, where $\Omega_P$ is the angular velocity at the centre of the
vortex. Models with density excess can exhibit many narrow parametric
instability bands while those with a concentrated vorticity source display
internal shear which significantly affects their stability. However, the
existence of these parametric instabilities may not necessarily prevent the
possibility of dust accumulation in vortices.
