Any hyperelliptic curve having base field characteristic dividing group order?
A subgroup of order equal to the basefield characteristic, a large prime?
Having hard DLP in that subgroup?
New elliptic-hyperelliptic cycle anyone?
SNARKs were introduced with QAP "intermediate" language to define computer-science and real-file problems as systems of equations "with just a single multiplication", widely known thanx ZCash. Recursive SNARKs suggest verifying "previous" snark-proof(s) as a part of the current problem instance, that could be illustrated with proving a correct transaction and "good" previous blockchain state resulting in good current state.
"A cycle of curves" like MNT4-MNT6 is the only known "complete" solution implementing groth16 SNARK. There is a good reason for no single-curve recursive snarks.
Group order is roughly twice-bitlength (a square of) the field a genus-2 hyperelliptic curve is defined over. It is not "slightly different" from the field characteristic anymore it was with elliptic curves. Potentially there is a rich structure hopefully admitting recursion with a single curve, or alternative cycles.
Hyperelliptic curves are not so popular. Group element is a pair of points on a genus-2 curve, so you need to handle "divisors" rather than points.
There was a project 2019-2020 that resulted in a manager claimed me to discover an MNT5 curve, made public on ZCash forum. That team did "proofs for knowledge" and "recurrent proofs" in their papers in a Springer journal and an IEEE conference.
The last point is the core question, the reason, and potentially a new idea.
MNT4-MNT6 elliptic curves pair is the only known "2-cycle": base field characteristic of the first equal to group order of the second, and vice versa. Informally, "splitting" the first property into two curves.
It seems feasible to have group order (Jacobian) to be a product of two primes, the base field characteristic $q$ and $(q+2)$ for a genus-2 (but not for elliptic) curve. At least, this is ok with the Hasse-Weil interval. This hopefully would mean avoiding "the largest prime divisor of the order of the Jacobian, is equal to the base field characteristic" (the wikipedia page suggested by Daniel S, thanx!) pre-condition of the known attack on anomalous curves. If confirmed, this might mean a "1-curve cycle" for zk-SNARKs.