Abstract
It is shown that gravitational fields invariant for a non Abelian 2-dimensional Lie algebra of Killing fields are parameterized either by solutions of a transcendental equation (the tortoise equation) or by solutions of a linear second order partial differential equation (the Laplace equation or the Darboux equation) on the plane. Those determined via Laplace or Darboux equations are exact nonlinear gravitational waves obeying to two nonlinear superposition laws.
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