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We consider the two matrix model with an even quartic potentialW(y)=y^4/4+alpha y^2/2 and an even polynomial potential V(x). The main resultof the paper is the formulation of a vector equilibrium problem for thelimiting mean density for the eigenvalues of one of the matrices M_1. Thevector equilibrium problem is defined for three measures, with external fieldson the first and third measures and an upper constraint on the second measure.The proof is based on a steepest descent analysis of a 4 x 4 matrix valuedRiemann-Hilbert problem that characterizes the correlation kernel for theeigenvalues of M_1. Our results generalize earlier results for the casealpha=0, where the external field on the third measure was not present.

The Hermitian two matrix model with an even quartic potential