  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 $V(z,u)$, $(z,u)\in {\mathbb R}^2\times {\mathbb R}$,
harmonic in $z$ (i.e. source-free), the trajectories of its associated
dynamical system $\ddot{z}(s)=-\nabla_z V(z(s),s)$ 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 $u\in {\mathbb R}$. The mathematical and physical
implications of this {\em polynomial EK conjecture}, as well as the
non-polynomial one, are discussed beyond their original scope.
