  We give an integral representation for the Kajihara's $q$-hypergeometric
series $W^{M,2}$. And we construct a $q$-difference system that corresponds to
this integral. This system is an extension of the variant of $q$-hypergeometric
equation of degree three, defined by Hatano-Matsunawa-Sato-Takemura. We show
that this system includes the $q$-Appell-Lauricella system as a degeneration.
