  The normal matrix model with algebraic potential has gained a lot of
attention recently, partially in virtue of its connection to several other
topics as quadrature domains, inverse potential problems and the Laplacian
growth.
  In this paper we consider the normal matrix model with cubic plus linear
potential. To regularize the model, we follow Elbau & Felder and introduce a
cut-off. In the large size limit, the eigenvalues of the model accumulate
uniformly within a certain domain $\Omega$ that we determine explicitly by
finding the rational parametrization of its boundary.
  We also study in details the mother body problem associated to $\Omega$. It
turns out that the mother body measure $\mu_*$ displays a novel phase
transition that we call the mother body phase transition: although $\partial
\Omega$ evolves analytically, the mother body measure undergoes a "one-cut to
three-cut" phase transition.
  To construct the mother body measure, we define a quadratic differential
$\varpi$ on the associated spectral curve, and embed $\mu_*$ into its critical
graph. Using deformation techniques for quadratic differentials, we are able to
get precise information on $\mu_*$. In particular, this allows us to determine
the phase diagram for the mother body phase transition explicitly.
  Following previous works of Bleher & Kuijlaars and Kuijlaars & L\'opez, we
consider multiple orthogonal polynomials associated with the model. Applying
the Deift-Zhou nonlinear steepest descent method to the associated
Riemann-Hilbert problem, we obtain asymptotic formulas for these polynomials.
Due to the presence of the linear term in the potential, there are no
rotational symmetries in the model. This makes the construction of the
associated $g$-functions significantly more involved, and the critical graph of
$\varpi$ becomes the key technical tool in this analysis as well.
