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We prove that a connected Riemannian manifold admitting a pair of non-trivial Einstein-Weyl structures $(g, \pmømega)$ with constant scalar curvature is either Einstein, or the dual field of $ømega$ is Killing. Next, let $(M^n, g)$ be a complete and connected Riemannian manifold of dimension at least $3$ admitting a pair of Einstein-Weyl structures $(g, \pmømega)$. Then the Einstein-Weyl vector field $E$ (dual to the $1$-form $ømega$) generates an infinitesimal harmonic transformation if and only if $E$ is Killing.