I have never heard of a law requiring anyone to maintain households. If you owe Alimony or back Child Support you can apply to the courts for deferred payment plans. If you currently have possession of the children of the woman who gave birth in who you pay, then simple research can help you discover your resources available to you.
There are community legal services that are available to you that can give you legal council free of charge. If this woman living in better financial statue then her children, there are laws that can help even the balance of family separate households.
If all fails you may have to reconsider your parental duties until you repay your dept in full. Life is only as hard as you allow it to be on you!!