  It is conjectured that a class of n-fold integral transformations
{I(alpha)|alpha in {C}} forms a mutually commutative family, namely, we have
I(alpha) I(beta)=I(beta) I(alpha) for all alpha, beta in {C}. The commutativity
of I(alpha) for the two-fold integral case is proved by using several summation
and transformation formulas for the basic hypergeometric series. An explicit
formula for the complete system of the eigenfunctions for n=3 is conjectured.
In this formula and in a partial result for n=4, it is observed that all the
eigenfunctions do not depend on the spectral parameter alpha of I(alpha).
