  The Matsumoto--Yor (MY) property of the generalized inverse Gaussian and
gamma distributions has many generalizations. As it was observed in (Letac and
Weso{\l}owski in Ann Probab 28:1371--1383, 2000) the natural framework for the
multivariate MY property is symmetric cones; however they prove their results
for the cone of symmetric positive definite real matrices only. In this paper,
we prove the converse to the symmetric cone-variate MY property, which extends
some earlier results. The smoothness assumption for the densities of respective
variables is reduced to the continuity only. This enhancement was possible due
to the new solution of a related functional equation for real functions defined
on symmetric cones.
