Motions of fluid particles advected by a vortex soliton are studied. In a reference frame which moves with the vortex soliton, particle motions are confined in a torus near the loop part of the vortex soliton for a wide range of three parameters that characterize the shape and strength of the vortex soliton. The transported volume is calculated numerically as a function of these parameters. The product of the volume and the translational velocity of the soliton provides the rate of transport. Using this quantity, the optimized shape of the soliton for the maximum rate of transport is considered. The torus is composed of groups of invariant surfaces around periodic trajectories. Similar phenomena are observed with the KAM tori for non-integrable Hamiltonian systems. To extract the essential mechanism of the transport properties, an ordinary differential equation model is proposed, which is named the ‘chopsticks model’. This model successfully explains the qualitative features of the transport.