The paper examines the phenomenon of steady-state motion for a string traveling with constant velocity along an invariant curve under gravity in a viscous medium. This technically important phenomenon has been known in the literature for about 120 years and may be applied in high-speed turbines, the textile industry, etc. The conditions for the phenomenon's existence are found. Concepts of two critical string velocities as well as sub, super, and hypercritical domains are introduced. The analytical solutions for the nonlinear differential equations and arbitrary constants for the general boundary conditions are found. The theoretical results are very close to the experimental ones.
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