HH-contact semi-Riemannian manifolds

We characterize HH-contact semi-Riemannian manifolds (i.e., contact semi-Riemannian manifolds whose Reeb vector field ξξ is harmonic) by the condition that ξξ is a Ricci eigenvector. We then investigate how HH-contact semi-Riemannian manifolds are related to some relevant geometric properties, like the Reeb vector field being an infinitesimal harmonic transformation or the contact semi-Riemannian structure being a contact Ricci soliton, and we determine to what extent known results for contact Riemannian manifolds remain valid in the contact Lorentzian case.