On regular algebraic surfaces of $\mathbb{R}^3$ with constant mean curvature

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We consider regular surfaces $M$ that are given as the zeros of a polynomial function $p : \mathbb{R}^3 \to \mathbb{R}$, where the gradient of $p$ vanishes nowhere. We assume that $M$ has non-zero constant mean curvature and prove that there exist only two examples of such surfaces, namely the sphere and the circular cylinder.