  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.
