Hi, thanks! In that sentence I was only talking about the translations and rotations of the plane as a group of invariances for the action of the two-body prob

Hi, thanks! In that sentence I was only talking about the translations and rotations of the plane as a group of invariances for the action of the two-body problem. This group is generated by one-parameter subgroups producing vertical translation, horizontal translation, and rotation about a particular point. Those are the "three degrees of freedom" I was counting. You're right about the correspondence from symmetries to conservation laws in general.