I can see there can be multiple interpretations of this question. I will be interested in an answer to any of them.
"quantum systems" could include
- Actual systems observed in experiment with quantum characteristics
- Actual systems that must exist in nature with quantum characteristics
- Conceptual models or scenarios within accepted quantum theory
- Speculative quantum models beyond accepted theory
"no eigenstates" could be because
- There is no Hamiltonian at all (I realise this stretches the grammar of the question slightly). For example maybe a system with no time aspect / no explicit time aspect? Or for any other reason.
- There is a Hamiltonian, but it has no eigenstates. Whether or not this is allowed would be even more interesting for me.