  The paper is divided into two parts. In the first part we lay down the
foundation for defining the joint annihilation-preservation-creation
decomposition of a finite family of, not necessarily commutative random
variables, and show that this decomposition is essentially unique. In the
second part we show that any two, not necessarily commutative, random variables
$X$ and $Y$, for which the vector space spanned by their annihilation,
preservation, and creation operators equipped with the bracket given by the
commutator, forms a Lie algebra, are equivalent, up to an invertible linear
transformation to two independent Meixner random variables with mixed
preservation operators. In particular if $X$ and $Y$ commute, then they are
equivalent, up to an invertible linear transformation to two independent
classic Meixner random variables. To show this we start with a small technical
condition called "non-degeneracy".
