On the rational monodromy-free potentials with sextic growth

We study the rational potentials $V(x)$, with sextic growth at infinity, such that the corresponding one-dimensional \Sch equation has no monodromy in the complex domain for all values of the spectral parameter. We investigate in detail the subclass of such potentials which can be constructed by the Darboux transformations from the well-known class of quasi-exactly solvable potentials $$V= x^6 - \nu x^2 +\frac{l(l+1)}{x^2}.$$ We show that, in contrast with the case of quadratic growth, there are monodromy-free potentials which have quasi-rational eigenfunctions, but which can not be given by this construction. We discuss the relations between the corresponding algebraic varieties, and present some elementary solutions of the Calogero-Moser problem in the external field with sextic potential.