Various solutions are displayed and analyzed (both analytically and numerically) of a recently-introduced many-body problem in the plane which includes both integrable and nonintegrable cases (depending on the values of the coupling constants); in particular the origin of certain periodic behaviors is explained. The light thereby shone on the connection among integrability and analyticity in (complex) time, as well as on the emergence of a chaotic behavior (in the guise of a sensitive dependance on the initial data) not associated with any local exponential divergence of trajectories in phase space, might illuminate interesting phenomena of more general validity than for the particular model considered herein.