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Second-Chern-Einstein metrics on 4-dimensional almost-Hermitian manifolds

We study 4-dimensional second-Chern-Einstein almost-Hermitian manifolds. In the compact case, we observe that under a certain hypothesis the Riemannian dual of the Lee form is a Killing vector field. We use that observation to describe 4-dimensional compact second-Chern-Einstein locally conformally symplectic manifolds and we give some examples of such manifolds. Finally, we study the second-Chern-Einstein problem on unimodular almost-abelian Lie algebras, classifying those that admit a left-invariant second-Chern-Einstein metric with a parallel non-zero Lee form.