Question on superconductors

can, at least in theory, there exist a class of superconductors that are NOT repelled by magnetic fields and have a London penetration depth of zero?

Or would such a thing be logically impossible and, if so, why?

If not logically impossible, is there a known example of such a superconductor and, if so, what is it?

If you define "superconductivity" as "zero resistance", then, yes, I believe so. If I'm not mistaken a pure, defect-free conductor has zero resistance but won't have the same magnetic properties. But the way "superconductivity" is usually defined, i.e. involving Cooper pairs, I think it implies those magnetic properties.

After looking on Wikipedia, it seems this distinction is usually made and "superconductivity" and "perfect conductivity" are not the same, although they both imply 0 resistance.