Two-component soliton systems and the Painlev\'e equations

We give an extension of the two-component KP hierarchy by considering additional time variables. We obtain the linear $2\times 2$ system by taking into consideration the hierarchy through a reduction procedure. The Lax pair of the Schlesinger system and the sixth Painlev\'e equation is given from this linear system. A unified approach to treat the other Painlev\'e equations from the usual two-component KP hierarchy is also considered.