  Let $\{\mathbb{P}_n\}_{n\ge 0}$ and $\{\mathbb{Q}_n\}_{n\ge 0}$ be two monic
polynomial systems in several variables satisfying the linear structure
relation $$\mathbb{Q}_n = \mathbb{P}_n + M_n \mathbb{P}_{n-1}, \quad n\ge 1,$$
where $M_n$ are constant matrices of proper size and $\mathbb{Q}_0 =
\mathbb{P}_0$. The aim of our work is twofold. First, if both polynomial
systems are orthogonal, characterize when that linear structure relation exists
in terms of their moment functionals. Second, if one of the two polynomial
systems is orthogonal, study when the other one is also orthogonal. Finally,
some illustrative examples are presented.
