  Spectra and magnetic properties of large spins $J$, placed into a crystal
electric field (CEF) of an arbitrary symmetry point group, are shown to change
drastically when $J$ changes by 1/2 or 1. At a fixed field symmetry and
configuration of its $N$ extrema situated at $p$-fold symmetry axis, physical
characteristics of the spin depend periodically on $J$ with the period equal to
$p$. The problem of the spectrum and eigenstates of the large spin $J$ is
equivalent to analogous problem for a scalar charged particle confined to a
sphere $S^2$ and placed into magnetic field of the monopole with the charge
$J$. This analogy as well as strong difference between close values of $J$
stems from the Berry's phase occurring in the problem. For energies close to
the extrema of the CEF, the problem can be formulated as Harper's equation on
the sphere. The $2J+1$-dimensional space of states is splitted into smaller
multiplets of classically degenerated states. These multiplets in turn are
splitted into submultiplets of states transforming according to specific
irreducible representations of the symmetry group determined by $J$ and $p$. We
classify possible configurations and corresponding spectra. Experimental
realizations of large spins in a symmetric environment are proposed and
physical effects observable in these systems are analyzed.
