Proves Vaisman solvmanifolds must be finite quotients of Kodaira-Thurston manifolds, implying LCK Einstein-Weyl structures do not exist.
We prove that every Vaisman solvmanifold has to be a finite quotient of a Kodaira-Thurston manifold. Consequently, we obtain that every completely solvable solvmanifold admitting a Vaisman structure has to be a Kodaira-Thurston manifold, and the non-existence of LCK Einstein-Weyl structures on solvmanifolds.
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Lucas H. S. Gomes (2025) studied this question.
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