automorphism group – What is the list of all the regular planar graphs with equivalent vertices?

Platonic solids, Archimedean solids, prisms and anti prism all have planar graphs where all the vertices are equivalent, or in other words, for any 2 vertices v1 and v2 there is an automorphism that maps v1 into v2.

What about the reverse problem: are the planar graphs for which all the vertices are equivalent only the ones corresponding to the Platonic solids, Archimedean solids, prisms and anti prism? If so, can it be proved easily or where can we find such a proof?