  We produce examples of solutions to the non-abelian gravitating vortex
equations, which are a dimensional reduction of the K\"aher-Yang-Mills- Higgs
equations. These are equations for a K\"ahler metric and a metric on a vector
bundle. We consider a symmetric situation on a sphere with a relationship
between the parameters involved (criticality), and perform a non-trivial
reduction of the problem to a system of ordinary differential equations on the
real line with complicated boundary conditions at infinity. This system
involves a parameter whose dependence on the volume of the K\"ahler metric is
non-explicit. We prove existence to this system using the method of continuity.
We then prove that the parameter can be varied to make sure that all possible
admissible volumes are attained.
