The mass Lagrangian coordinate associated with the velocity field of the gas for a nonstationary drift-flux model of a gas-liquid mixture is introduced here. If the mass transfer is neglected, this results in a quite simple structure of the governing equations and permits to integrate one of the equations independently on the others. The simplified model, describing the horizontal two-phase flow for high Froude number is considered and some explicit solutions to this model in the mass Lagrangian coordinates are obtained.