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Working in the framework of Lizorkin distributions, and hence without any boundedness or decay hypothesis on $u$, we give a complete classification of the solutions: $u$ solves $P(i\\nabla)u=0$ if and only if $\\hat{u}$ is a multi-layer distribution on $S^{d-1}$ of order at most $N$. Alternatively, $u$ solves $P(i\\nabla)u=0$ if and only if $(1+\\Delta)^{N+1}u=0$ if $P$ satisfies a flatness condition. 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Working in the framework of Lizorkin distributions, and hence without any boundedness or decay hypothesis on $u$, we give a complete classification of the solutions: $u$ solves $P(i\\nabla)u=0$ if and only if $\\hat{u}$ is a multi-layer distribution on $S^{d-1}$ of order at most $N$. Alternatively, $u$ solves $P(i\\nabla)u=0$ if and only if $(1+\\Delta)^{N+1}u=0$ if $P$ satisfies a flatness condition. 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