  The algebraic structures underlying quasi-exact solvability for spin 1/2
Hamiltonians in one dimension are studied in detail. Necessary and sufficient
conditions for a matrix second-order differential operator preserving a space
of wave functions with polynomial components to be equivalent to a \sch\
operator are found. Systematic simplifications of these conditions are
analyzed, and are then applied to the construction of several new examples of
multi-parameter QES spin 1/2 Hamiltonians in one dimension.
