  We find a new property in $W^2$-conformal gravity in spherical symmetry. We
demonstrate that the charge of the electromagnetic field varies with respect to
the partial scaling symmetry (conformal transformations in subspaces of a
spacetime) in the conformal gravity. We find that the electric and magnetic
charges can vanish or appear via partial conformal transformations for the
on-shell
  configurations in the conformal gravity. We find a solution of a Yang-Mills
SU(2)-charged black hole in the conformal gravity, and further demonstrate that
the SU(2) charge has similar properties with electromagnetic magnetic charges
in the conformal gravity. We demonstrate how to imbed the electromagnetic and
Yang-Mills SU(2) gauge field into the conformal gravity via such a symmetry. We
make a brief discussion about the possibility to extend this argument to the
Einstein gravity.
