A method is proposed for investigating plasma oscillations in axially symmetric systems with a longitudinal magnetic field, which can be used for an arbitrary ratio of the Larmor radius of charged particles to the radius of the system. An anisotropic ion cyclotron instability and a modified negative-mass instability are analysed with this method. The analysis was performed for a system with parameters similar to those of the Ogra-3 device. It is shown, in particular, that in the case of a modified negative-mass instability, the threshold density for oscillations of the first azimuthal mode has a lower value than for the zero mode. This result corresponds to experimental data.