  The dispersionless KP hierarchy is considered from the point of view of the
twistor formalism. A set of explicit additional symmetries is characterized and
its action on the solutions of the twistor equations is studied. A method for
dealing with the twistor equations by taking advantage of hodograph type
equations is proposed. This method is applied for determining the orbits of
solutions satisfying reduction constraints of Gelfand--Dikii type under the
action of additional symmetries.
