Abstract
In this work we consider the three-dimensional solvable Lie group
denoted by $Sol_{3}$, equipped with any left-invariant metric,
either Lorentzian or Riemannian. The existence of non-trivial (i.e., not
Einstein) Ricci solitons on both Lorentzian and Riemannian
three-dimensional solvable Lie group $Sol_{3}$ is proved. Moreover,
we show that they are not gradient Ricci solitons.