  We apply a recently developed method to exactly solve the $\Phi^3$ matrix
model with covariance of a two-dimensional theory, also known as regularised
Kontsevich model. Its correlation functions collectively describe graphs on a
multi-punctured 2-sphere. We show how Ward-Takahashi identities and
Schwinger-Dyson equations lead in a special large-$\mathcal{N}$ limit to
integral equations that we solve exactly for all correlation functions.
Remarkably, these functions are analytic in the $\Phi^3$ coupling constant,
although bounds on individual graphs justify only Borel summability.
  The solved model arises from noncommutative field theory in a special limit
of strong deformation parameter. The limit defines ordinary 2D Schwinger
functions which, however, do not satisfy reflection positivity.
