<b>Abstract</b>— <p>An analytical method for finding the natural frequencies and eigenfunctions of transverse vibrations of a four-component fuel element (rod) for the designed NEPTUNE pulse reactor is proposed. As a model, a rod with a piecewise constant material density in the longitudinal direction is used. The frequency equation is explicitly obtained via Krylov functions, and the natural frequencies of the first eight modes are determined using a numerical and analytical method. To find higher modes, an exponential analytical extrapolation is proposed. Analytical expressions for eigenfunctions providing a good agreement with numerical calculations in the ANSYS environment are obtained.</p>