Can FOC be applied to single phase Capacitor start Capacitor run motor?.

What is the question, the subject line and message body are unrelated.

If controlled, single phase ACI machines are controlled using V/Hz principle. With capacitor run motor you might need to put a proper filter between inverter and machine as the capacitor is probably not designed to handle high frequency currents. That is if you connect the machine with two wires only.

This is an awful amount of trouble. Much better solution is to replace the machine with ordinary three-phase one and drive it with ordinary inverter.

You could connect auxiliary winding to inverter (through capacitor or directly). But the FOC scheme becomes quite complicated if you use three phase inverter and main phase connected between two inverter phases and auxilary phase connected on one side to main phase and on the other side to remaining inverter phase.

System becomes much nicer to handle if you drive each phase with its own full bridge. Any you don't need any capacitors. Again it is usually cheaper to change the machine.

Fyi, in many applications, even in appliances where single phase IM were used, now mfrs are moving out of it to 3 phase motors for higher efficiency in operation and control. If you are still interested, you may refer to some text books on motors and drives. I don't have any reference to suggest.