  $W$-representation is a miraculous possibility to define a non-perturbative
(exact) partition function as an exponential action of somehow integrated Ward
identities on unity. It is well known for numerous eigenvalue matrix models
when the relevant operators are of a kind of $W$-operators: for the Hermitian
matrix model with the Virasoro constraints, it is a $W_3$-like operator, and so
on. We extend this statement to the monomial generalized Kontsevich models
(GKM), where the new feature is the appearance of an ordered P-exponential for
the set of non-commuting operators of different gradings.
