Framed $4$-valent Graph Minor Theory II: Special Minors and New Examples

In the present paper, we proceed the study of framed $4$-graph minor theory initiated in ``Framed $4$-valent Graph Minor Theory I. Intoduction. Planarity Criterion ‘’ and justify the planarity theorem for arbitrary framed 4-graphs; besides, we prove analogous results for embeddability in the projective plane.