Reyes Nozaleda, DanielTempesta, PiergiulioTondo, Giorgio2023-06-222023-06-222023-040938-897410.1007/s00332-022-09877-8https://hdl.handle.net/20.500.14352/73182© 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.engAtribución 3.0 Españajournal articlehttp://dx.doi.org/10.1007/s00332-022-09877-8https://link.springer.com/open access51-73HaantjesSystemsFísica-Modelos matemáticosFísica matemática