Dynamic converters use an idler motor to inductively create the third leg. Both systems will cost some money.
But if you've got a spare three-phase motor - HP rating as big as the one in the lathe or somewhat larger - lying around it takes very little money.
Mount it solidly to something. Hook up power to it (through a motor contactor or suitable switch for motor loads) on two of its legs, a big RUN capacator (or several in parallel) between one of those and the third leg. (They're mostly acting as start capacitors but run capacitors are rated for continuous duty so you don't have to unhook them once the motor is running - and they DO help during run, too.) Hook the three wires from the motor also to the input lines on your lathe.
Turn on the motor's switch with the lathe off, wait until it's up to full speed, and turn on the lathe. You can switch off both at once or lathe first.
Nothing to it.
(Hanging capacitors across the lathe motor in the same way and leaving out the idler motor will get it to start and run. But you'll have to reduce your cutting rates to about a third of normal to keep the motor from overheating unless you use a LOT of capacitors.)