Self-similar analytical nonlinear solutions to the hydrostatic Boussinesq equations are derived which describe unbalanced inertial pulsations of anticyclonic lens-like circular vortices in stably stratified rotating fluid. Any steady axisymmetric solution for a finite-volume anticyclonic vortex in the reduced-gravity approximation is shown to correspond to a set of time-periodic solutions with the amplitude of pulsations being within a range limited by the intensity of the stationary vortex. These solutions represent an extension of previous reduced-gravity analytical pulson solutions of particular forms with spatially uniform divergence of horizontal velocity oscillating in time within the vortex volume. In the self-similar form the pulson solution describes the expansion and contraction of a vortex which maintains the same spatial structure in the Lagrangian coordinates.