  The explicit solvability of quantum superintegrable systems is due to
symmetry, but the symmetry is often "hidden". The symmetry generators of 2nd
order superintegrable systems in 2 dimensions close under commutation to define
quadratic algebras, a generalization of Lie algebras. Distinct systems on
constant curvature spaces are related by geometric limits, induced by
generalized In\"on\"u-Wigner Lie algebra contractions of the symmetry algebras
of the underlying spaces. These have physical/geometric implications, such as
the Askey scheme for hypergeometric orthogonal polynomials. However, the limits
have no satisfactory Lie algebra contraction interpretations for underlying
spaces with 1- or 0-dimensional Lie algebras. We show that these systems can be
best understood by transforming them to Laplace conformally superintegrable
systems, with flat space conformal symmetry group ${\rm SO}(4,{\mathbb C})$,
and using ideas introduced in the 1894 thesis of B\^ocher to study separable
solutions of the wave equation in terms of roots of quadratic forms. We show
that B\^ocher's prescription for coalescing roots of these forms induces
contractions of the conformal algebra $\mathfrak{so}(4,{\mathbb C})$ to itself
and yields a mechanism for classifying all Helmholtz superintegrable systems
and their limits. In the paper [Acta Polytechnica, to appear,
arXiv:1510.09067], we announced our main findings. This paper provides the
proofs and more details.
