  We construct a class of one-dimensional Lie-algebraic problems based on sl(2)
where the spectrum in the algebraic sector has a dynamical symmetry E -> - E.
All 2j+1 eigenfunctions in the algebraic sector are paired, and inside each
pair are related to each other by simple analytic continuation x -> ix, except
the zero mode appearing if j is integer. At j-> infinity the energy of the
highest level in the algebraic sector can be calculated by virtue of the
quasiclassical expansion, while the energy of the ground state can be
calculated as a weak coupling expansion. The both series coincide identically.
