We study a semilinear elliptic system modeling the physical system strings and antistrings in cosmology under the boundary condition of the symmetric vacuum (the nontopological type). We construct solutions with the representation having precise informations on the asymptotic behaviors near infinity for arbitrary location of strings and antistrings satisfying
$\[$1\leqslant M-N<\tfrac{1}{4\pi G}$$
, where M and N are the total string number and the total antistring number respectively, and G is the gravitational constant. The asymptotic properties, in particular, are completely different to the solutions under the boundary condition of the asymmetric vacuum (the topological type) constructed previously by Y. Yang [Phys. Rev. Lett. 80 (1999), 26–29]. We also compute the total magnetic flux, total energy and the total Gaussian curvature of our solutions.
