The dressing method, symmetries, and invariant solutions

The dressing method associates to a given nonlinear equation for q, a Riemann-Hilbert problem or a problem uniquely determined in terms of certain inverse data ƒ. Thus it generates a map from solutions of a linear system of PDEs (that for ƒ) to a nonlinear system of PDEs (that for q). We show that the corresponding tangent map can be expressed in closed form. Hence, symmetries and invariant solutions of ƒ induce symmetries and invariant solutions for q. The procedure can be used to charaterize solutions of Painlevé equations.