By investigating a harmonically confined and periodically driven particle system with spin-orbit coupling (SOC) and a specific controlled parameter,we demonstrate an exactly solvable two-level model with a complete set of spin-motion entangled Schr(o)dinger kitten (or cat) states.In the undriven case,application of a modulation resonance results in the exact stationary states.We show a decoherence-averse effect of SOC and implement a transparent coherent control by exchanging positions of the probability-density wavepackets to create transitions between the different degenerate ground states.The expected energy consisting of quantum and continuous parts is derived,and the energy deviations caused by the exchange operations are much less than the quantum gap.The results could be directly extended to a weakly coupled single-particle chain for transparently encoding spin-orbit qubits via the robust spin-motion entangled degenerate ground states.