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Let us consider Beauville's Integrable System. So, we live on $\mathbb{P}^1$. There is the moduli space of matrices $M_r(d)/\mathrm{PGL}(r)$ with polynomial entries of degree less than or equal $d$. According to results of Donagi and Markman if we impose condition on the leading coefficient $A_d$ of $M \in M_{d}(R)/\mathrm{PGL}(r)$ i.e. $A_d$ is regular nilpotent then there is a canonical section of certain kind of the quotient $M_r(d)/\mathrm{PGL}(r)$ over the locus where $A_d$ is regular. Do you know some references on description of this locus in the case when leading coefficient is not necessarily regular nilpotent? I'm interested in the case of general nilpotent.

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