  We report Zeeman infra-red spectroscopy of electronic-nuclear levels of
$^5$I$_8 \rightarrow ^5$I$_7$ transitions of Ho$^{3+}$ in the C$_{\rm
4v}$(F$^-$) centre in CaF$_2$ with the magnetic field along the $\langle
111\rangle$ direction of the crystal. Transitions to the lowest $^5$I$_7$
state, an isolated electronic doublet, and the next group of states, a
pseudo-quadruplet consisting of a doublet and two nearby singlets, exhibit
strongly non-linear Zeeman splittings and intensity variations. Simulated
spectra based upon a crystal-field analysis give an excellent approximation to
the data, illustrating the strong predictive ability of the parametrised
crystal-field approach. Anti-crossings in the hyperfine splittings, the basis
of quantum information storage in rare-earth doped insulating dielectrics, are
also predicted.
