  We study the functional equation \[ \sum_{i=1}^mf_i(b_ix+c_iy)=
\sum_{k=1}^nu_k(y)v_k(x) \] with $x,y\in\mathbb{R}^d$ and $b_i,c_i\in
{GL}(d,\mathbb{R})$, both in the classical context of continuous complex-valued
functions and in the framework of complex-valued Schwartz distributions, where
these equations are properly introduced in two different ways. The solution
sets are, typically, exponential polynomials and, in some particular cases,
related to so called characterization problem of the normal distribution in
Probability Theory, they reduce to ordinary polynomials.
