  In this paper we study regular cosmic string solutions of the non-Abelian
Higgs model coupled with the Einstein gravity. In order to do that, we
constructed a set of coupled differential ordinary equation. Because there is
no closed solution for this set of equations, we solve it numerically. The
solutions that we are interested in asymptote to a flat space-time with a
planar angle deficit. This model under consideration present two bosonic
sectors, besides the non-Abelian gauge one, coupled minimally with the
gravitational fields. The two bosonic sectors may present a direct coupling,
which plays an important role on the behavior of the matter and gauge fields
and also on the behavior on the geometry of the spacetime. We explicitly
analyze the behaviors of the energy density and planar angle deficit as
function of the energy scale where the gauge symmetry is spontaneously broken
and the coupling interaction between the bosonic sectors.
