We consider a uniform ellipsoid of potential vorticity (PV), where we exploit analytical solutions derived for a balanced model at the second order in the Rossby number, the next order to quasi-geostrophic (QG) theory, the so-called QG+1 model. We consider this vortex in the presence of an external background shear flow, acting as a proxy for the effect of external vortices. For the QG model the system depends on four parameters, the height-to-width aspect ratio of the vortex,
$h/r$
, as well as three parameters characterising the background flow, the strain rate,
$\gamma$
, the ratio of the background rotation rate to the strain,
$\beta$
, and the angle from which the flow is applied,
$\theta$
. However, the QG+1 model also depends on the PV, as well as the Prandtl ratio,
$f/N$
(
$f$
and
$N$
are the Coriolis and buoyancy frequencies, respectively). For QG and QG+1 we determine equilibria for different values of the background flow parameters for increasing values of the imposed strain rate up to the critical strain rate,
$\gamma _c$
, beyond which equilibria do not exist. We also compute the linear stability of this vortex to second-order modes, determining the marginal strain
$\gamma _m$
at which ellipsoidal instability erupts. The results show that for QG+1 the most resilient cyclonic ellipsoids are slightly prolate, while anticyclonic ellipsoids tend to be more oblate. The highest values of
$\gamma _m$
occur as
$\beta \to 1$
. For large values of
$f/N$
, changes in the marginal strain rates occur, stabilising anticyclonic ellipsoids while destabilising cyclonic ellipsoids.