This paper proposes a new modular multilevel converter (MMC) configuration as a medium-voltage drive for variable-speed applications incorporating symmetrical six-phase machines. The proposed topology employs six MMC phase-legs feeding two isolated groups of three-phase machine windings, each with 60° spatial phase-displacement. A novel concept of sharing one capacitor between each pair of adjacent-arm submodules (SMs) of MMC phase-legs, while feeding machine windings in a spatial phase-opposition, is realized through a new integrated SM arrangement. The integrated SM allows the shared capacitor to absorb and release the same energy amount in a consecutive switching scheme, where the capacitor is experiencing both charging and discharging arm currents, one after another. This results in a limited voltage variation across the SM shared capacitor, independent of the operating frequency. Also, the proposed approach allows the MMC to utilize half the number of the SM capacitors, compared to a traditional MMC topology, while further diminishes the SM capacitance requirement, reducing the volume of the MMC system and its stored energy. The proposed configuration can efficiently operate at near zero frequency, therefore a machine speed-range from zero speed to the rated speed is possible under rated torque operating condition. The proposed MMC topology is elucidated in detail, and its effective performance is verified using simulation.