  A simple theoretical solution to the design of a control field that generates
complete population transfer from an initial state, via $N$ nondegenerate
intermediate states, to one arbitrary member of $M$ ($M\leq N$) degenerate
states is constructed. The full control field exploits an $(M+N-1)$-node null
adiabatic state, created by designing the relative phases and amplitudes of the
component fields that together make up the full field. The solution found is
universal in the sense that it does not depend on the exact number of the
unwanted degenerate states or their properties. The results obtained suggest
that a class of multi-level quantum systems with degenerate states can be
completely controllable, even under extremely strong constraints, e.g., never
populating a Hilbert subspace that is only a few dimensions smaller than the
whole Hilbert space.
