3-phase power to run lathe
You likely have what is known as a three-phase rotary converter. A quick google search will present uncounted returns for various designs. A friend of mine had one that required a kick-start in order to operate. He had a flywheel attached to the 220v motor which when switched on would not rotate until he struck it with his foot. Once up to speed, it then provided to the lathe motor the necessary power configuration to operate.

I would not begin to suggest that your configuration matches his, as his was home-spun and created without access to the internet! The links I've found have variations which may be challenging to match to yours.

One option you have is to replace the rotary converter with what is called a VFD converter. It is going to be a simpler device, with the added benefit of speed control as well as likely a more reliable bundle for your application. Unfortunately, it is an additional purchase, while you already have the devices on hand for the rotary phase converter.