Classification of motions of a relativistic string with massive ends with linearizable boundary conditions

We classified all motions (world surfaces) of a relativistic string with massive ends, for which equations of motion and boundary conditions can be linearized through a natural parametrization of the end's trajectories. These motions can be represented as Fourier series with eigenfunctions of some generalization of the Sturm–Liouville problem. Completeness of a set of these eigenfunctions in class $C$ is proved. It is shown that in $2+1$ and $3+1$-dimensional Minkowski spaces all these motions reduce to an uniform rotation of a straight string or some such spatially coincident strings (world surface is helicoid). In spaces with higher dimensionality other non-trivial motions of the investigated type are possible.