Low-frequency longitudinal (POGO) oscillations of liquid launch vehicles is a phenomenon inherent to almost all liquid rockets. POGO oscillations of launch vehicles can lead to various emergencies: damages of the rocket structure and liquid propellant propulsion system, unacceptable malfunctions of the rocket control system. The use of liquid-propellant rocket engines with an oxidizer-rich staged combustion cycle for the first stage of launch vehicles can introduce a number of features into the POGO stability analysis. First of all, in this case, longitudinal vibrations of launch vehicles can occur due to the low-frequency instability of a liquid propulsion system at frequencies associated with the dynamics of the circuit of the turbopump-gas generator-gas duct. Another feature of these engines is the manifestation of a significant maximum of the module of the engine dynamic pressure gain in the low frequency range (up to 10 Hz), which can lead to POGO instability of the launch vehicle even in the initial part of its flight with significant values of the rocket structure generalized masses for the lower modes of launch vehicle natural vibrations. To predict the POGO stability of the currently designed Cyclone-4M two-staged launch vehicle, the mathematical model of the low-frequency dynamics of the "propulsion system-rocket structure" system has been developed. The model describes the interac