  We use the Kontsevich--Miwa transform to relate the Virasoro constraints on
integrable hierarchies with the David-Distler-Kawai formalism of
gravity-coupled conformal theories. The derivation relies on evaluating the
energy-momentum tensor on the hierarchy at special values of the spectral
parameter. We thus obtain in the Kontsevich parametrization the `master
equations' which implement the Virasoro constraints and at the same time
coincide with null-vector decoupling equations in an `auxiliary' conformal
field theory on the complex plane of the spectral parameter. This gives the
operators their gravitational scaling dimensions (for one out of four
possibilities to choose signs), with the $\alpha_+$ being equal to the
background charge $Q$ of an abstract $bc$ system underlying the structure of
the Virasoro constraints. The formalism also generalizes to the $N$-KdV
hierarchies.
