Conditions ensuring regularity under iterated Volterra map

Determine explicit conditions on an initial triple of polynomials (P(x), Q(x), R(x)) as defined by equation (2.1) such that the parameter u from equation (2.2) remains nonzero for all iterates under the Volterra map Vg, thereby guaranteeing that every triple produced by repeated iterations is regular (i.e., the map is well-defined at each step).

Background

The Volterra map Vg is defined via the Lax equation L(x)M(x)=M(x)L(x) and acts birationally on triples (P,Q,R) of polynomials with a parameter u given by a linear combination of the coefficients (equation (2.2)). The map is only defined when u ≠ 0, and such triples are termed regular. While an inverse mapping exists when u ≠ 0, the authors note that regularity of iterates is not guaranteed for an arbitrary starting triple. Identifying sufficient (or necessary and sufficient) conditions that ensure u stays nonzero under all iterations would clarify when bi-infinite dynamics are well-defined.