  We obtain the most general type B 3-fold supersymmetry by solving directly
the intertwining relation. We then show that it is a necessary and sufficient
condition for a second-order linear differential operator to have three
linearly independent local analytic solutions. We find that there are eight
linearly independent non-trivial linear differential operators of this kind. As
a by-product, we find new quasi-solvable second-order operators preserving a
monomial or polynomial subspace, one in type B, two in type C, and four in type
X_2, all of which have been missed in the existing literature. In addition, we
show that type A, type B, and type C 3-fold supersymmetries are connected
continuously via one parameter. A few new quasi-solvable models are also
presented.
