You can either use the Vertex Weight Edit modifier or, more simply in this case, the Vertex Weight Mix modifier:
Which will let you subtract the band group from the circle group:
Ensure 'Use Modifier Stack' is checked in the particle system..
Use the options in the armature modifier: "Display modifier in edit mode" and "Adjust edit cage to the modifier result".
In complement, concerned bones are generally the one used for nearly connected vertices, you can see them in the properties panel:
Rigify is a quite complicated rig, in which a lot of bones are "control" or "machinary" bones, so they don't deform the mesh (they don't have the "deform" option activated, so no associated vertex group is created when parenting).
All and the only bones who deform the mesh are stored into a generally hidden bone layer, and have the "DEF" suffix on their ...
You need to parent the mesh to the armature first to setup the weights, this is done with Lock Object Modes checked.
In Object Mode, select the mesh, Shift select armature, then Ctrl + P to parent and choose Armature Deform then select any sub-option, like With Automatic Weights.
You should now be able to Ctrl + left mouse on the bones in weight paint mode ...