  Saddle points of a vector logarithmic energy with a vector polynomial
external field on the plane constitute the vector critical measures, a notion
that finds a natural motivation in several branches of analysis. We study in
depth the case of measures $\vec \mu=(\mu_1, \mu_2,\mu_3)$ when the mutual
interaction comprises both attracting and repelling forces.
  For arbitrary vector polynomial external fields we establish general
structural results about critical measures, such as their characterization in
terms of an algebraic equation solved by an appropriate combination of their
Cauchy transforms, and the symmetry properties (or the S-properties) exhibited
by such measures. In consequence, we conclude that vector critical measures are
supported on a finite number of analytic arcs, that are trajectories of a
quadratic differential globally defined on a three-sheeted Riemann surface. The
complete description of the so-called critical graph for such a differential is
the key to the construction of the critical measures.
  We illustrate these connections studying in depth for a one-parameter family
of critical measures under the action of a cubic external field. This choice is
motivated by the asymptotic analysis of a family of (non-hermitian) multiple
orthogonal polynomials, that is subject of a forthcoming paper. Here we compute
explicitly the Riemann surface and the corresponding quadratic differential,
and analyze the dynamics of its critical graph as a function of the parameter,
giving a detailed description of the occurring phase transitions. When
projected back to the complex plane, this construction gives us the complete
family of vector critical measures, that in this context turn out to be vector
equilibrium measures.
