We consider non-axisymmetric oscillations of a thin straight magnetic tube.
The tube is modeled by a magnetic cylinder with the background quantities discontinuous at the cylinder boundary.
The unperturbed state is axisymmetric and the background quantities are independent on the radial coordinate both inside and outside the cylinder.
However they do depend on the coordinate along the cylinder.
The magnetic field lines are assumed to be frozen in dense plasma at the cylinder bases.
To simplify the problem we use the cold plasma approximation.
The assumption that the cylinder is thin enables us to fine the dependence of perturbed quantities on the radial coordinate.
After that, using the boundary conditions at the cylinder boundary we reduce the problem
to a simple Sturm-Leuville problem for a second order ordinary differential equation.
We solve this problem in two particular cases and apply the obtained results to oscillations of cylindrical prominence fibrils
