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© The Author(s) 2023 We wish to thank the anonymous referees for many useful observations. This work has been partly supported by the research project PGC2018-094898-B-I00, MICINN, Spain, and by the Severo Ochoa Programme for Centres of Excellence in R & D (CEX2019-000904-S), Ministerio de Ciencia, Innovacion y Universidades y Agencia Estatal de Investigacion, Spain. D. R. N. acknowledges the financial support of EXINA S.L.P. T. and G. T. are members of the Gruppo Nazionale di Fisica Matematica (GNFM)., The theory of generalized Nijenhuis torsions, which extends the classical notions due to Nijenhuis and Haantjes, offers new tools for the study of normal forms of operator fields. We prove a general result ensuring that, given a family of commuting operator fields whose generalized Nijenhuis torsion of level m vanishes, there exists a local chart where all operators can be simultaneously block-diagonalized. We also introduce the notion of generalized Haantjes algebra, consisting of operators with a vanishing higher-level torsion, as a new algebraic structure naturally generalizing standard Haantjes algebras., Ministerio de Ciencia e Innovación (MICINN), Centros de Excelencia Severo Ochoa (MICINN), EXINA S.L.P. T., Depto. de Física Teórica, Fac. de Ciencias Físicas, TRUE, pub

© 2022 Elsevier The research of D. R. N. has been supported by the Severo Ochoa Programme for Centres of Excellence in R&D (CEX2019-000904-S), Ministerio de Ciencia, Innovacion y Universidades, Spain. The research of P. T. has been supported by the research project PGC2018-094898-B-I00, Ministerio de Ciencia, Innovacion y Universidades, Spain, and by the Severo Ochoa Programme for Centres of Excellence in R&D (CEX2019-000904-S), Ministerio de Ciencia, Innovacion y Universidades, Spain. The research of G. T. has been supported by the research project FRA2020-2021, Universitadegli Studi di Trieste, Italy. P. T. is member of Gruppo Nazionale di Fisica Matematica (GNFM) of INDAM., We show that the theory of classical Hamiltonian systems admitting separating variables can be formulated in the context of (omega, H ) structures. They are symplectic manifolds en-dowed with a compatible Haantjes algebra H , namely an algebra of (1,1)tensor fields with vanishing Haantjes torsion. A special class of coordinates, called Darboux-Haantjes coor-dinates, will be constructed from the Haantjes algebras associated with a separable sys-tem. These coordinates enable the additive separation of variables of the corresponding Hamilton-Jacobi equation. We shall prove that a multiseparable system admits as many omega H structures as sepa-ration coordinate systems. In particular, we will show that a large class of multiseparable, superintegrable systems, including the Smorodinsky-Winternitz systems and some physi-cally relevant systems with three degrees of freedom, possesses multiple Haantjes struc-tures. (C) 2021 Published by Elsevier B.V., Ministerio de Ciencia e Innovación (MICINN), Centros de Excelencia Severo Ochoa (MICINN), Universitá degli Studi di Trieste, Depto. de Física Teórica, Fac. de Ciencias Físicas, TRUE, pub