Variations of Saffman's robot: two mechanisms of locomotion?

We study two mechanisms of locomotion of a body in an inviscid fluid, which take place without the shedding of vorticity; we consider two simple examples of robots which are able to move along a straight trajectory. The first one consists of a sphere with an internal moving material point (actuator); this illustrates \emph{the recoil locomotion}. The second robot represents a sphere moving along a thin rigid rod, this is aimed at illustrating \emph{the deformational locomotion}. The latter appears since this motion of a sphere can be seen as a `soliton' deformation, moving along the rod. The equations of motion for both robots are the same, while the intervals of variables are different. The first robot was introduced by \cite{Saffman}, who also wrote that he was unable to find any exact solution for the deformational locomotion: our paper fills this gap. Some previous papers emphasize that, in the cases of locomotion, the deformations must be not axisymmetric. We consider only axisymmetric examples, which expands the range of the involved possibilities. The simple construction of presented robots allows us to operate with the exact solutions only, which can play a `reference' role. Our aim is to analyse the main notions and terminology, used in this high-impact research area.