Exigent is not really a nomenclature the way it's being used here. They're Exalted. "Exigent" refers more to the bureaucratic policy that defined how they came into existence. So, there is no "Solar Exigent."
To answer the OP: The Incarnae can't make any more Exalts, with or without the Exigence. (And they didn't need the Exigence in the first place.)