We analyse in a systematic way the (non-) compact n-dimensional Einstein–Weyl spaces equipped with a cohomogeneity-one metric. In that context, with no compactness hypothesis for the manifold on which lives the Einstein–Weyl structure, we prove that, as soon as the ( n−1)-dimensional space is a homogeneous reductive Riemannian space with a unimodular group of left-acting isometries G: • there exists a Gauduchon gauge such that the Weyl-form is co-closed and its dual is a Killing vector for the metric; • in that gauge, a simple constraint on the conformal scalar curvature holds; • a non-exact Einstein–Weyl structure may exist only if the ( n−1)-dimensional homogeneous space G/ H contains a non-trivial subgroup H′ that commutes with the isotropy subgroup H; • the extra isometry due to this Killing vector corresponds to the right-action of one of the generators of the algebra of the subgroup H′. The first two results are well known when the Einstein–Weyl structure lives on a compact manifold, but our analysis gives the first hints on the enlargement of the symmetry due to the Einstein–Weyl constraint. We also prove that the subclass with G compact, a one-dimensional subgroup H′ and the ( n−2)-dimensional space G/( H× H′) being an arbitrary compact symmetric Kähler coset space, corresponds to n-dimensional Riemannian locally conformally Kähler metrics. The explicit family of structures of cohomogeneity-one under SU( n/2) being, thanks to our results, invariant under U(1)× SU( n/2), it coincides with the one first studied by Madsen; our analysis allows us to prove most of his conjectures.