Constructions of helicoidal minimal surfaces and minimal annuli in $\widetilde{E(2)}$

In this article, we construct two one-parameter families of properly embedded minimal surfaces in a three-dimensional Lie group $\widetilde{E(2)}$, which is the universal covering of the group of rigid motions of Euclidean plane endowed with a left-invariant Riemannian metric. The first one can be seen as a family of helicoids, while the second one is a family of catenoidal minimal surfaces. The main tool that we use for the construction of these surfaces is a Weierstrass-type representation introduced by Meeks, Mira, P\'erez and Ros for minimal surfaces in Lie groups of dimension three. In the end, we study the limit of the catenoidal minimal surfaces. As an application of this limit case, we get a new proof of a half-space theorem for minimal surfaces in $\widetilde{E(2)}$.