  This paper is a continuation of our paper [Med. J. Math 19, Article number:
31 (2022)] in which we extended the notion of generalized Drazin-Riesz
invertible operators to closed operators. We establish here, results relating
the notion of closed generalized Drazin-Riesz invertibility with the theory of
$C_{0}$-semigroups. Firstly, we generalize results obtained in the bounded case
[1] to the context of closed operators. Secondly, we investigate when an
infinitesimal generator $A$ of a given $C_{0}$-semigroup is closed generalized
Drazin-Riesz invertible. An application to $C_{0}$-groups and abstract second
order differential equations is proposed, and an example of a $C_{0}$-group
with closed generalized Drazin-Riesz invertible infinitesimal generator is
given.
