Analytical solutions are found to describe a water parcel motion in both warm and cold eddies. An f plane, reduced gravity model, with upper layer thickness and velocity fields respectively Gaussian and linear in space, was reduced to the eight-dimensional system of nonlinear coupled ordinary differential equations which describe time evolution of differential kinematic parameters and eddy shape parameters. Analytical solutions for steady rotating modes for warm eddies and for cold eddies are discussed. For warm eddies the rigidly rotating ellipse mode solution corresponds to the ''rodon'' solution for lenslike warm eddies. For cold eddies the corresponding rigidly rotating mode solution exhibits triangular path with abrupt direction change. Elliptical paths for warm eddies and triangular paths for cold eddies were observed in Western Pacific Ocean Circulation Study III drifter tracking data, which may be interpreted to be a possible manifestation of the rotating mode solutions predicted theoretically. ¿ American Geophysical Union 1992