  The one-dimensional Schrodinger equation for the potential $x^6+\alpha x^2
+l(l+1)/x^2$ has many interesting properties. For certain values of the
parameters l and alpha the equation is in turn supersymmetric (Witten),
quasi-exactly solvable (Turbiner), and it also appears in Lipatov's approach to
high energy QCD. In this paper we signal some further curious features of these
theories, namely novel spectral equivalences with particular second- and
third-order differential equations. These relationships are obtained via a
recently-observed connection between the theories of ordinary differential
equations and integrable models. Generalised supersymmetry transformations
acting at the quasi-exactly solvable points are also pointed out, and an
efficient numerical procedure for the study of these and related problems is
described. Finally we generalise slightly and then prove a conjecture due to
Bessis, Zinn-Justin, Bender and Boettcher, concerning the reality of the
spectra of certain PT-symmetric quantum-mechanical systems.
