The Kitaev model is a beautiful example of frustrated interactions giving rise to deep and unexpected phenomena. In particular, its classical version has remarkable properties stemming from exponentially large ground state degeneracy. Here, we present a study of magnetic clusters with spin-$S$ moments coupled by Kitaev interactions. We focus on two cluster geometries -- the Kitaev square and the Kitaev tetrahedron -- that allow us to explicitly enumerate all classical ground states. In both cases, the classical ground state space (CGSS) is large and self-intersecting, with non-manifold character. The Kitaev square has a CGSS of four intersecting circles that can be embedded in four dimensions. The tetrahedron CGSS consists of eight spheres embedded in six dimensions. In the semi-classical large-$S$ limit, we argue for effective low energy descriptions in terms of a single particle moving on these non-manifold spaces. Remarkably, at low energies, the particle is tied down in bound states formed around singularities at self-intersection points. In the language of spins, the low energy physics is determined by a distinct set of states that lies well below other eigenstates. These correspond to `Cartesian states, a special class of classical ground states that are constructed from dimer covers of the underlying lattice. They completely determine the low energy physics despite being a small subset of the classical ground state space. This provides an example of order by singularity, where state selection becomes stronger upon approaching the classical limit.
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