  Recently it has been argued that the ground state of high density QCD is
likely be a combination of the CFL-phase along with condensation of the K^0
field. This state spontaneously breaks a global U(1)_Y symmetry, therefore one
would expect the formation of U(1)_Y global strings. We discuss the core
structure of these strings and demonstrate that under some conditions the
global U(1)_Y symmetry may not be restored inside the string, in contrast with
the standard expectations. Instead, K^+ condensation occurs inside the core of
the string if a relevant parameter \cos\theta_K = \mkzero^2/\mu_eff^2 is larger
than some critical value \theta_K > \theta_crit. If this phenomenon happens,
the U(1)_Y strings become superconducting and may considerably influence the
magnetic properties of dense quark matter, in particular in neutron stars.
