  Boundary equations for the relativistic string with masses at ends are
formulated in terms of geometrical invariants of world trajectories of masses
at the string ends. In the three--dimensional Minkowski space $E^1_2$, there
are two invariants of that sort, the curvature $K$ and torsion $\kappa$.
Curvatures of trajectories of the string ends with masses are always constant,
$K_i = \gamma/m_i (i =1,2,)$, whereas torsions $\kappa_i(\tau)$ obey a system
of differential equations with deviating arguments. For these equations with
periodic $\kappa_i(\tau+n l)=\kappa(\tau)$, constants of motion are obtained
(part I) and exact solutions are presented (part II) for periods $l$ and $2l$
where $l$ is the string length in the plane of parameters $\tau$ and $\sigma \
(\sigma_1 = 0, \sigma_2 =l)$.
