Can a 3 phase motor run on single phase?
A 3-phase motor, once rotating, can keep rotating even if one phase is lost (wire cut), but the current in the remaining 2 phases will rise and create overheating and vibrations. Generally the thermal relay includes a protection against motor operation in single-phase. This is generally a 'must' in motor protection strategy.
If you want to connect a 3-phase motor to a single phase power supply, then you have several solutions:
Professional: use a VFD with a single phase input and convert to 3-phase output, this is generally available for low power VFD (few horsepower).
Professional: apply a specially wound transformer that also mimic a 3-phase system from a single phase power supply. Expect difficulties to source and high price.
DIY: calculate a capacitor that will be connected between the remaining phases, such to apply it a phase shift and to mimic a 3-phase system.
By operating a 3-phase motor from a single-phase supply is most often done by people with a hobby workshop and want to be able to operate a bit of 3-phase equipment, they perhaps bought cheaply. In other words a lath or wood cutting machine which they perhaps use very occasionally, so can't afford the proper 3-phase VFD equipment, just want to do a Heath Robinson home constructed adaption.