  In supersymmetric quantum mechanics, shape invariance is a sufficient
condition for solvability. We show that all conventional additive shape
invariant superpotentials that are independent of $\hbar$ obey two partial
differential equations. One of these is equivalent to the one-dimensional Euler
equation expressing momentum conservation for inviscid fluid flow, and it is
closed by the other. We solve these equations, generate the set of all
conventional shape invariant superpotentials, and show that there are no others
in this category. We then develop an algorithm for generating all additive
shape invariant superpotentials including those that depend on $\hbar$
explicitly.
