Ehlers-Kundt conjecture is a physical assertion about the fundamental role of plane waves for the description of gravitational waves. Mathematically, it becomes equivalent to a problem on the Euclidean plane ${mathbb R}^2$ with a very simple formulation in Classical Mechanics: given a non-necessarily autonomous potential $V(z,u)$, $(z,u)in {mathbb R}^2times {mathbb R}$, harmonic in $z$ (i.e. source-free), the trajectories of its associated dynamical system $ddot{z}(s)=- abla_z V(z(s),s)$ are complete (they live eternally) if and only if $V(z,u)$ is a polynomial in $z$ of degree at most $2$ (so that $V$ is a standard mathematical idealization of vacuum). Here, the conjecture is solved in the significative case that $V$ is bounded polynomially in $z$ for finite values of $uin {mathbb R}$. The mathematical and physical implications of this {em polynomial EK conjecture}, as well as the non-polynomial one, are discussed beyond their original scope.
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