  According to the generalized Polya theorem, the Gaussian distribution on the
real line is characterized by the property of equidistribution of a monomial
and a linear form of independent identically distributed random variables. We
give a complete description of a-adic solenoids for which an analog of this
theorem is true. The proof of the main theorem is reduced to solving some
functional equation in the class of continuous positive definite functions on
the character group of an a-adic solenoid
