It is demonstrated that, in each of two problems which involve physica
l symmetry, the appropriate eigenvalue equation factorizes into two eq
uations, one of which corresponds to solutions which are even function
s of the independent variable, and the other to solutions which are od
d functions. The first situation involves a two-point boundary value p
roblem and a linear differential equation. The second involves the eig
envalue problem for a matrix which is symmetric about both of its diag
nols.