How can this 3-phase PM Synchronous AC servo motor operate with single-phase?
It is not the motor that can work on either single-phase or 3-phase, it is the control unit. The control unit supplies 3-phase to the motor in all cases. The advantage of using 3-phase is that using predominately 3-phase equipment in a factory make it easier to balance the total factory load among the three phases. There is also some advantage in reducing the harmonic content of the load current drawn from the mains. There may be some convenience advantage to having three smaller power conductors rather than two larger conductors.

For both 3-phase and single-phase input, mains is rectified and filtered to provide DC that is then converted to controlled 3-phase AC. For a controller that will accept either single-phase or 3-phase, the rectifier needs to have a high enough current rating to supply the required power using only 4 of the six diodes. The filter capacitor needs to be adequate for un-filtered DC with higher ripple voltage.