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This work is intended to contextualize and enhance certain well-studiedrelationships between Hessenberg varieties and the Toda lattice, therebybuilding on the results of Kostant, Peterson, and others. One such relationshipis the fact that every Lagrangian leaf in the Toda lattice is compactified by asuitable choice of Hessenberg variety. It is then natural to imagine the Todalattice as extending to an appropriate union of Hessenberg varieties.  We fix a simply-connected complex semisimple linear algebraic group $G$ andrestrict our attention to a particular family of Hessenberg varieties, a familythat includes the Peterson variety and all Toda leaf compactifications. Thetotal space of this family, $X(H_0)$, is shown to be a Poisson variety with acompletely integrable system defined in terms of Mishchenko--Fomenkopolynomials. This leads to a natural embedding of completely integrable systemsfrom the Toda lattice to $X(H_0)$. We also show $X(H_0)$ to have an open densesymplectic leaf isomorphic to $G/Z \times S_{\text{reg}}$, where $Z$ is thecentre of $G$ and $S_{\text{reg}}$ is a regular Slodowy slice in the Liealgebra of $G$. This allows us to invoke results about integrable systems on$G\times S_{\text{reg}}$, as developed by Rayan and the second author. Lastly,we witness some implications of our work for the geometry of regular Hessenbergvarieties.

Hessenberg varieties, Slodowy slices, and integrable systems