On a problem with a boundary condition of the second kind, a complex-valued coefficient, and a spectral parameter in the other boundary condition

We study a classical problem that arises in the analysis of natural vibrations of a loaded string with a free endpoint. We assume that the coefficient occurring in the boundary condition of the third kind with a spectral parameter instead of a physical parameter can take complex values. We discuss the traditional aspects of the completeness, minimality, and basis property of the system of root functions. Special attention is paid to the structure of root subspaces.