  Distributional equation is an important tool in the characterization theory
because many characteristic properties of distributions can be transferred to
such equations. Using a novel and natural approach, we retreat a remarkable
distributional equation whose corresponding functional equation in terms of
Laplace-Stieltjes transform is of the Poincare type. The necessary and
sufficient conditions for the equation to have a unique distributional solution
with finite variance are provided. This complements the previous results which
involve at most the mean of the distributional solution. Besides, more general
distributional (or functional) equations are investigated as well.
