**The problem of existence of****elementary (= quasi-exact****) multiplets of bound states is discussed.****Background: It is easy to assign a potential V(f) to any****given****elementary wavefunction f.**

**Idea: For the doublet of solutions f and g the two potentials must coincide, V(f)=V(g).****Conclusion: Our doublet- (or multiplet-) solvable class {V} will depend on our "solution form" {f}.****We discuss the feasibility of this constuction and its algebraic background and analytic applications.**

**You may click on the ps formatted****full text****description of these results as published in****J. Math. Phys. 33 (1992) 2785 - 94.****An alternative Riccati-like development of the same idea may be found in the Ushveridze's book (1994).**