  A definition of $d$--dimensional $n$--Meixner random vectors is given first.
This definition involves the commutators of their semi--quantum operators.
After that we will focus on the $1$-Meixner random vectors, and derive a system
of $d$ partial differential equations satisfied by their Laplace transform. We
provide a set of necessary conditions for this system to be integrable. We use
these conditions to give a complete characterization of all non--degenerate
three--dimensional $1$--Meixner random vectors. It must be mentioned that the
three--dimensional case produces the first example in which the components of a
$1$--Meixner random vector cannot be reduced, via an injective linear
transformation, to three independent classic Meixner random variables.
