Matrix model partition function by a single constraint

In the recent study of Virasoro action on characters, we discovered that it gets especially simple for peculiar linear combinations of the Virasoro operators: particular harmonics of ŵ -operators. In this letter, we demonstrate that even more is true: a single w -constraint is sufficient to uniquely specify the partition functions provided one assumes that it is a power series in time-variables. This substitutes the previous specifications in terms of two requirements: either a string equation imposed on the KP/Toda τ -function or a pair of Virasoro generators. This mysterious single -entry definition holds for a variety of theories, including Hermitian and complex matrix models, and also matrix models with external matrix: the unitary and cubic Kontsevich models. In these cases, it is equivalent to W-representation and is closely related to super integrability. However, a similar single equation that completely determines the partition function exists also in the case of the generalized Kontsevich model (GKM) with potential of higher degree, when the constraint algebra is a larger W -algebra, and neither W-representation, nor superintegrability are understood well enough.