Can you run a single phase motor, with run capacitor, rated 220V with a 3 phase supply (220V delta) and without the run capacitor. I mean supplying each wires coming out of the motor with 3 phase 220V instead of using a single phase 220V plus the run capacitor. Since you can start a 3 phase motor with a single phase supply by adding a capacitor can you do it the other way around?
Yes you can run the 1-phase, 220V motor from a line-line connection on a 3-phase, 220V delta. You would still need the start capacitor. Also, make sure case grounding is properly maintained as applicable (detailed wiring drawing will show).
If you are asking if you can supply the start capacitor leg with one of the other line-line delta voltages (for out-of-phase voltage) IN PLACE of the start capacitor then you will have to ask the manufacturer. Using another line-line would provide a full \$±120°\$ shift whereas using the start capacitor and the same line-line voltage would be something less (e.g. \$-90°<\theta<+90°\$).
Do you have a detailed wiring/winding drawing for the motor?