Cyclic splittings of pro-$\mathcal{C}$ groups

In this paper we proved a pro-$\mathcal{C}$ version of the Rips-Sela Theorems on splittings as an amalgamated free product or HNN-extension over a cyclic subgroup, where $\mathcal{C}$ is a class of finite groups closed for subgroups, quotients, finite direct products and extensions with abelian kernel.