Referring to South Dakota Law: see below for details of scenario.

No. Since she is incapacitated a judge needs to appoint a Guardian. She can no longer grant POA due to incapacity. You have no rights to her financial stuff. If there is no health care proxy, then South Dakota next of kin laws would apply in an emergency. In the following order, SD defaults for next of kin are:
• Spouse
• Adult child
• Parent
• Adult sibling
• Grandparent or adult grandchild
• Aunt or uncle or adult niece or nephew
• Adult cousin
• Close friend

May I ask why the three grown children relinquished the Power of Attorney?

It seems odd that the three grown children still want to decide on what to do financially and health wise for Mother, so why give up the legal piece of paper POA that state they can???

Depending on State Laws, it could be possible that the Court will say that Mom needs someone to represent her rights, thus the Court could appoint someone from the Court to handle Mom financials and health decisions. It is too late for Mom to appoint the three children back onto the POA since she is incapacitated.

Seek the advice of an Elder Law Attorney.