  We present a set of conditions which, if satisfied, provide for a complete
asymptotic analysis of random matrices with source term containing two distinct
eigenvalues. These conditions are shown to be equivalent to the existence of a
particular algebraic curve. For the case of a quartic external field, the curve
in question is proven to exist, yielding precise asymptotic information about
the limiting mean density of eigenvalues, as well as bulk and edge
universality.
