ABSTRACT Recent observations of stellar occultations have revealed rings of particles around non-planetary bodies of the Solar system. These bodies are irregular and can be modelled by ellipsoids. In the context of numerical integrations for the study of studying the region close to these ellipsoidal bodies, it is known that the use of geometric initial conditions is necessary when the central object is significantly oblatened. In this paper, we show that for elongated bodies there is also a need for the adaptation of the initial velocity ($\nu _{C_{22}}$) so that equatorial periodic orbits of the first kind around this body have smaller radial variations since the circular Keplerian velocity produces a high oscillating eccentricity and radial variation. We describe an empirical method to obtain the velocity $\nu _{C_{22}}$ of a set of simulations where we vary the physical parameters of the central body. With the obtained data, developed empirical equations that allow the calculation of the orbital eccentricity, the initial velocity and an adapted Kepler’s Third Law as a function of the ellipticity coefficient and the semimajor axis. In addition, we identify an important change in the location of the primary body in relation to the elliptical orbit. In the cases of the orbits with minimal radial variation found in our study, the body starts to occupy the centre of the elliptical orbit. Finally, we include the rotation of the central body in the studied systems and analyse its implications for the dynamics of these orbits of low radial variation.