2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/209682We establish a characterization of coagulation-fragmentation processes, such that the induced birth and death processes depicting the total number of groups at time $t\ge 0$ are time homogeneous. Based on this, we provide a characterization of mean-field Gibbs coagulation-fragmentation models, which extends the one derived by Hendriks et al. As a by- product of our results, the class of solvable models is widened and a question posed by N. Berestycki and Pitman is answered, under restriction to mean-field models.30pages.This is the final version that contains a few important changes in exposition implied by referees remarks and questions. The paper will be published in J. of Statistical PhysicsProbabilityCombinatorics60J27, 60J35, 05A18, 82C23text