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We propose a more direct approach to constructing differential operators thatpreserve polynomial subspaces than the one based on considering elements of theenveloping algebra of sl(2). This approach is used here to construct newexactly solvable and quasi-exactly solvable quantum Hamiltonians on the linewhich are not Lie-algebraic. It is also applied to generate potentials withmultiple algebraic sectors. We discuss two illustrative examples of these twoapplications: an interesting generalization of the Lam\'e potential whichposses four algebraic sectors, and a quasi-exactly solvable deformation of theMorse potential which is not Lie-algebraic.Comment: 17 pages, 3 figure

Quasi-exact solvability and the direct approach to invariant subspaces