The small platonic solids

The tetrahedron (3.3.3) has a unique EP-embedding.

Likewise, the octahedron (3.3.3.3) has only a single EP-embedding without coincident vertices.

The cube (4.4.4) has a three-fold infinity of EP-embeddings: all parallelepipeds with equal sides serves as valid embeddings. If you enforce the embedding to have (at least) T symmetry, then only the classical embedding of the cube remains.