Picking up on the point about the maximum frequency that a nominal 50 Hz motor can be driven I've outline my experience with my lathe.
I have a Chipmaster lathe which I've modified by removing the variator but have retained the original 3kW 4 pole 1428 RPM motor which is configured in Delta for 240V working. I drive the motor from a 3 phase 7kW VFD which I operate from a 240v single phase supply. The VFD is programmed to have a max output frequency of 100Hz that spins the motor at 2800rpm and I have fitted a motor pulley to gives a spindle speed of 2200rpm. This works fine and produces sufficient torque at low speed for most of the turning I do whilst providing adequate higher speeds.
However the Chipmaster is capable of running its spindle at 3000 rpm so to try and get this I re-programmed the VFD max frequency to 150Hz but still couldn't achieve much more than 2500rpm which I was due to the motor's back emf and the supply voltage.
I next tried a 6 pole 980 rpm 3kW motor, fitted a bigger pulley but still couldn't get this motor to run at much more than twice its design speed confirming that the back emf appears to be the issue.
Therefore to get the higher speed I either have to drive a delta connected 240V motor from a 440V VFD or find a 110v motor and drive it from a 240V motor.
Clive