  We study field theoretical models for cosmic strings with flat directions in
curved space-time. More precisely, we consider minimal models with semilocal,
axionic and tachyonic strings, respectively. In flat space-time, the string
solutions of these models have a flat direction, i.e., a uniparametric family
of configurations with the same energy exists which is associated to a zero
mode. We prove that the zero mode survives coupling to gravity, and study the
role of the flat direction when coupling the strings to gravity. Even though
the total energy of the solution is the same, and thus the global properties of
the family of solutions remains unchanged, the energy density, and therefore
the gravitational properties, are different. The local structure of the
solutions depends strongly on the value of the parameter describing the flat
direction; for example, for supermassive strings, the value of the free
parameter can determine the size of the universe.
