This work considers the case of degenerate vibronic coupling of N states all at one single;point. For this case we first derived the extended Single-state Born-Oppenheimer equation [M. Baer and R. Englman, Chem. Phys. Lett. 265, 105 (1997)] and then showed that such a system, Like the two-state system, can be described in terms of a single phase factor and is characterized by a well-defined geometrical phase.