  In [8], Arveson proved that a $1$-parameter decomposable product system is
isomorphic to the product system of a CCR flow. We show that the structure of a
generic decomposable product system, over higher dimensional cones, modulo
twists by multipliers, is given by an isometric representation $V$ of the cone
and a certain $2$-cocycle for $V$. Moreover, we compute the space of
$2$-cocycles for shift semigroups associated to transitive actions of a higher
dimensional cone.
