Exact analytical solutions are found to describe f plane time-dependent, elliptical warm-core rings where the interface intersects the surface along the periphery. The space variables can be eliminated to reduce the problem to a system of differential equations in time. The motion of the center of mass is resolved and subtracted. Small departures from circular shape have three intrinsic frequencies: two are inertial and superinertial, while the third is a low-frequency mode that corresponds to a slow rotation of the elliptical eddy without change in shape. An exact solution for steadily rotating elliptical eddies of finite eccentricity (named Rodons) is also found and discussed. Comparison with elliptical warm-core Gulf Stream rings shows that this low-frequency mode may explain their clockwise rotation. The solution also shows the existence of a periodically reversing deformation field that, combined with a reasonable amount of mixing, would result in an efficient homogenization of the water contained in the ring. An exact solution that corresponds to the pulsation of a circular eddy has also been found.