Galaxy velocity mapsoften show the typical pattern of a rotating disk, consistent with the dynamical model where emitters rotate in circular orbits around the galactic center. The simplest template used to fit these maps consists in the rotating disk model (RDM) where the amplitude of circular velocities is fixed by the observed velocity profile along the kinematic axis. A more sophisticated template is the rotating tilted-ring model (RTRM) that takes into account the presence of warps and allows a radius-dependent orientation of the kinematic axis. In both cases, axisymmetry is assumed and residuals between the observed and the model velocity fields are interpreted as noncircular motions. We show that if a galaxy is not axisymmetric, there is an intrinsic degeneracy between a rotational and a radial velocity field. We then introduce a new galaxy template, the radial ellipse model (REM), that is not axisymmetric and has a purely radial velocity field with an amplitude that is correlated with the major axis of the ellipse. We show that best fits to the observed two-dimensional velocity fields of 28 galaxies extracted from the THINGS sample with both the REM and the RDM give residuals with similar amplitudes, where the REM residuals trace nonradial motions. Best fits obtained with the RTRM, because of its larger number of free parameters, give the smallest residuals: however, we argue that this does not necessarily imply that the RTRM gives the most accurate representation of a galaxy velocity field. Instead, we show that this method is not able to disentangle between circular and radial motions for the case of nonaxisymmetric systems. We then discuss a refinement of the REM, able to describe the properties of a more heterogeneous velocity field where circular and radial motions are respectively predominant at small and large distances from the galaxy center. Finally, we consider the physical motivation of the REM, and discuss how the interpretation of galactic dynamics changes if one assumes that the main component of a galaxy velocity field is modeled as a RDM/RTRM or as a REM.