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Vector solitary waves are nonlinear waves of coupled polarizations that propagate with constant velocity and shape. In mechanics, they hold the potential to control locomotion, mitigate shocks and transfer information, among other functionalities. Recently, such elastic waves were numerically observed in compressible rubber-like laminates. Here, we conduct numerical experiments to characterize the possible vector solitary waves in these laminates, and expose a new type of waves whose amplitude and velocity oscillate periodically without dispersing in time. This oscillation is a manifestation of a periodic transfer of energy between the two wave polarizations, which we consider as internal mode of the solitary wave. We find that the vector solitary waves propagate faster at higher amplitudes, and determine a lower bound for their velocity. We describe a procedure for identifying which initial strains generate such vector solitary waves. This procedure also enables an additional classification between tensile and compressive solitary waves, according to the way that the axial strain changes as the waves propagate.