Is it legal to run 3 phase motor in single phase?

Completely legal since it wont work at all. Or if it does depending on the windings you will get a slow spin with a loud humming sound. Not practical.

Yes, sort of. Experts on the CR4 forum report (1) that you can (poorly) run a three-phase motor by adding a capacitor connected to the third phase line. I've not tried that, but I can tell you that efficiently running 3-phase motors from single-phase AC power is VERY common. For example, if you have a modern washing machine, it no doubt uses a 3-phase motor.

In the industrial environment, three-phase motor drive-electronics are called VFD, or variable-frequency drives (2). VFD electronics packages include first, a rectifier bridge to get DC power from the AC power line, and second, a set of six IGBT switches (3) to create three pulse-width-modulated (PWM) sine waves 120-degree apart, as 3-phase power for the motor.

The huge advantage of creating 3-phase power this way rather than simply using 3-phase AC available at the wall, comes from the variable frequency aspect. For example, you can start the motor slowly and smoothly ramp up its speed, rather than presenting it with full-speed power, creating a loud thump.

In the case of a washing machine, you can make the motor run smoothly slowly back and forth, then when you're ready, ramp it up for spin dry. For the engineering inclined, I've added a link to a 32-page Infineon brochure (4) about designing three-phase drives.

It's not about legality. It's about practicality. You need a phase converter to make it work. A static phase converter uses a capacitor to start the motor on single phase, and it continues to run on single phase with only 2/3 the power output. A rotor phase converter includes a motor to start and produce the third leg so you get a complete three phase power.

You can run small 3-phase motors up to 5 Hp with a single to 3-phase converter. Without it your 3-phase motors won't run on single phase power supply.

3 phase motor needs 3 phase power supply..u cannt run with 1 phase..