Dynamical properties of singular Lagrangian systems differ from those of classical Lagrangians of the form $L=T-V$. Even less is known about symmetries and conservation laws of such Lagrangians and of their corresponding actions. In this article we study symmetries and conservation laws of a concrete singular Lagrangian system interesting in physics. We solve the problem of determining all point symmetries of the Lagrangian and of its Euler-Lagrange form, i.e. of the action. It is known that every point...