Oscillator spacetimes are Ricci solitons

We consider the four-dimensional oscillator group, equipped with a well-known one parameter family of left-invariant Lorentzian metrics, which includes the bi-invariant one (Gadea & Oubina, 1999). In a suitable system of global coordinates, the Ricci soliton equation for these metrics translates into a system of partial differential equations. Solving such system, we prove that all these metrics are Ricci solitons. In particular, the bi-invariant metric on the oscillator group gives rise to infinitely many Ricci solitons (and so, also to Yamabe solitons).