Nonlinear equations invariant under the Poincaré, similitude, and conformal groups in two-dimensional space-time

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Abstract
All realizations of the Lie algebras p(1,1), sim(1,1), and conf(1,1) are classified under the action of the group of local diffeomorphisms of R3. The result is used to obtain all second‐order scalar differential equations, invariant under the corresponding Poincaré, similitude, and conformal groups. The invariant equations are, in general, nonlinear, and the requirement of linearity turns out to be very restrictive. Group invariant solutions of some of the conformally invariant equations are obtained either by quadratures or by a linearizing transformation.