  Schroedinger equation with potentials of the Kratzer plus polynomial type
(say, quartic V(r) = A r^4 +B r^3 + C r^2+D r + F/r + G/r^2 etc) is considered.
A new method of exact construction of some of its bound states is then
proposed. it is based on the Taylor series terminated rigorously after N+1
terms at specific couplings and energies. This enables us to find the exact,
complete and compact unperturbed solution of the Magyari's N+2 coupled and
nonlinear algebraic conditions of the termination in the strong-coupling regime
with G \to \infty. Next, at G < \infty, we adapt the Rayleigh-Schroedinger
perturbation theory and define the bound states via an innovated, triple
perturbation series. In tests we show that all the correction terms appear in
integer arithmetics and remain, therefore, exact.
