Integrable equation of state for noisy cosmic string

It is argued that, independently of the detailed (thermal or more general) noise spectrum of the microscopic extrinsic excitations that can be expected on an ordinary cosmic string, their effect can be taken into account at a macroscopic level by replacing the standard isotropic Goto-Nambu-type string model by the nondegenerate string model characterized by an equation of state of the nondispersive "fixed determinant" type, with the effective surface stress-energy tensor satisfying ${\left(T_{}^{\nu }{}_{\nu }{}^{}\right)}^2-T_{}^{\mu }{}_{\nu }{}^{}T_{}^{\nu }{}_{\mu }{}^{}=2\mathrm{}{T}_0^2$, where $T_0$ is a constant representing the null-state limit of the string tension $T$, whose product with the energy density $U$ of the string is thereby held fixed: $TU=T_0^2$. It is shown that this equation of state has the special property of giving rise (in a flat background) to explicitly integrable dynamical equations.