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We consider generalisations of the elliptic Calogero--Moser systemsassociated to complex crystallographic groups in accordance to\cite{EFMV11ecm}. In our previous work \cite{Argyres:2021iws}, we proposedthese systems as candidates for Seiberg--Witten integrable systems of certainSCFTs. Here we examine that proposal for complex crystallographic groups ofrank one. Geometrically, this means considering elliptic curves $T^2$ with$\Z_m$-symmetries, $m=2,3,4,6$, and Poisson deformations of the orbifolds$(T^2\times\mathbb{C})/\Z_m$. The $m=2$ case was studied in\cite{Argyres:2021iws}, while $m=3,4,6$ correspond to Seiberg--Wittenintegrable systems for the rank 1 Minahan--Nemeshansky SCFTs of type$E_{6,7,8}$. This allows us to describe the corresponding elliptic fibrationsand the Seiberg--Witten differential in a compact elegant form. This approachalso produces quantum spectral curves for these SCFTs, which are given byFuchsian ODEs with special properties.

Complex crystallographic reflection groups and Seiberg-Witten integrable systems: rank 1 case