A320 Captain reported while on an ILS approach receiving a TCAS RA for unknown traffic resulting in the flight crew taking evasive action.

Narrative

Established on the ils or loc rwy xx approach to zzz. We pass zzzzz waypoint with 5100 ft. with heading 264 then the procedure says to descent to 4400ft towards zzzzz1 waypoint. First officer and I saw something that did not look like and regular airplane; it was small and it had south bound trajectory.First we had a ta and then the airplane started a descent to 4400 which we stopped with a push to level off at 5000ft. Then we had a ra advisory 'monitor vertical speed' which we comply with; being level off.Then we had the 'clear of conflict' and continue with the approach to rwy xx at zzz.During all time we saw this flying small plane flying and crossing in front of us.