Polyhedra are a standard math-art topic, but the Kepler-Poinsot solids and the infinite Petrie-Coxeter polyhedra are less emphasized. Their combination is even entirely new, and so it happened a new regular polyhedron, of infinite Petrie-Coxeter and Kepler-Poinsot type, was recently discovered. The present paper explores this case and two more: a tetrahedral, octahedral and an icosahedral symmetry case. It provides an example of an infinite Kepler-Poinsot solid in each case. It discusses their construction and, if possible, their generalized Euler-Cayley-formula.