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 Painlev\'{e} 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.