AbstractElliptic and hyperbolic geopotential solutions, for a homogeneous distribution of potential vorticity (PV), are obtained via PV inversion in geophysical vortices. The flow in the axisymmetrical three-dimensional vortices is steady and horizontal, where the centripetal acceleration plus the Coriolis acceleration equals the pressure anomaly gradient term (gradient wind or cyclo-geostrophic balance). It is found that the family of geopotential solutions in the vortex interior is completely parameterized by the PV density in the vortex and the squared aspect ratio between the horizontal and vertical semi-axes of the ellipsoidal or hyperbolic geopotential surfaces. Thus, the PV inversion task consists of obtaining, via solution of algebraic cubic equations, the absolute vertical vorticity and vertical stratification as a function of PV and aspect ratio. It is found that there is always a critical aspect ratio, which depends on PV, beyond which the PV inversion solutions are multi-valued. The complete vorticity and stratification solutions for the different regions in the PV and aspect ratio space are obtained and analysed with emphasis on the inertial and static instability of the vortex flow.