Converse characterization of diagonal-divisor Toda solutions

Determine what additional global rigidity hypothesis, beyond equality of the fundamental Plücker-form cone divisors, is sufficient to ensure that a unitary integral curve arises from a single spherical developing pair through a constant transformation of the principal curve, and hence that the associated Toda solution is reduced.

Background

The paper proves that every reduced solution generated by a cone spherical metric with divisor D has the diagonal cone-divisor tuple (D,…,D), but it also constructs an SU(3) counterexample showing that the converse fails: equal fundamental ramification divisors do not force reducedness. Reducedness is a global condition requiring the associated unitary integral curve to be a constant transformation of the principal curve composed with one spherical developing pair.

The authors identify a possible sufficient condition in type A2—namely, that the projective image be a conic—but leave open the broader question of what additional hypothesis yields a converse, particularly a global classification of unitary integral curves with equal fundamental ramification divisors.