Gauge Transformations and Weak Lax Equation

We consider several integrable systems from a standpoint of the SL(2,R) invariant gauge theory. In the Drinfeld-Sokorov gauge, we get a one parameter family of nonlinear equations from zero curvature conditions. For each value of the parameter the equation is described by weak Lax equations. It is transformed to a set of coupled equations which pass the Painlevé test and are integrable for any integer values of the parameter. Performing successive gauge transformations (the Miura transformations) on the system of equations we obtain a series of nonlinear equations.