  Symmetry based selection rules are developed providing minimal criteria for
the existence of two-channel Kondo interactions between conduction electrons
and the low energy degrees of freedom on U$^{4+}$ and Ce$^{3+}$ in a metal
host, assuming that the underlying microscopics are regulated by the Anderson
Hamiltonian. An additional dynamic selection rule is imposed on Ce$^{3+}$ ions.
The selection rules restrict the two-channel quadrupolar Kondo effect to
U$^{4+}$ ions in cubic, tetragonal, and hexagonal symmetry. For hexagonal and
tetragonal symmetry, the Kondo effect for a U$^{4+}$ ion will always be
quadrupolar. The selection rules for Ce$^{3+}$ ions restrict the two-channel
magnetic Kondo effect to one of three possible doublet ionic ground states in
hexagonal symmetry and the lone doublet ionic ground state in cubic symmetry.
The dynamical selection rule apparently excludes two-channel Kondo behavior for
Yb$^{3+}$ ions.
