  We present a conditionally exactly solvable singular potential for the
one-dimensional Schr\"odinger equation which involves the exactly solvable
inverse square root potential. Each of the two fundamental solutions that
compose the general solution of the problem is given by a linear combination
with non-constant coefficients of two confluent hypergeometric functions.
Discussing the bound-state wave functions vanishing both at infinity and in the
origin, we derive the exact equation for the energy spectrum which is written
using two Hermite functions of non-integer order. In specific auxiliary
variables this equation becomes a mathematical equation that does not refer to
a specific physical context discussed. In the two-dimensional space of these
auxiliary variables the roots of this equation draw a countable infinite set of
open curves with hyperbolic asymptotes. We present an analytic description of
these curves by a transcendental algebraic equation for the involved variables.
The intersections of the curves thus constructed with a certain cubic curve
provide a highly accurate description of the energy spectrum.
