Let C ⊂≠ Reg be a non-empty class (of regular cardinals). Then the logic has additional nice properties: it has the homogeneous model existence property.
@article{1371, title = {On $\lambda$ strong homogeneity existence for cofinality logic}, journal = {CUBO, A Mathematical Journal}, volume = {13}, year = {2011}, language = {en}, url = {http://dml.mathdoc.fr/item/1371} }
Shelah, Saharon. On λ strong homogeneity existence for cofinality logic. CUBO, A Mathematical Journal, Tome 13 (2011) . http://gdmltest.u-ga.fr/item/1371/