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<figure><img src="image_24.png" /></figure>

**Authors' contributions** AFS carried out most of the developments presented in this work, including: development and improvement on the symbolic software, analytical and numerical developments, analysis of the different cases, comparisons with perturbative solutions, results outputs and figure production. CT carried out the first steps of the research, developed the different normal forms, wrote the first draft of the paper and supervised the work. AV produced the first code version of *MORFE_Symbolic*, participated in the choice of the examples analysed, helped in producing the final version of the codes, and proposed numerous improvements in the presentation, the analytical developments and the symbolic processing. All authors read and approved the final manuscript.

**Supplementary Material** The source code of *MORFE_Symbolic* is available at github.com/adfs-dev/MORFE_Sym
bolic. A long-term archived snapshot is preserved by Software Heritage at archive.softwareheritage.org/swh:1:dir:094e48
d7fa354e07782fbbd45c734274cc8c71de;origin=https://github.com/MORFEproject/MORFE_Symbolic;visit=swh:1
snp:d7e9212f20989b423c83344f9fc10fac2d6adc9a;anchor=swh:1:rev:0372006b860e668846998f1fodf43a0cff96489d.
The associated dataset is available at 10.5281/zenodo.15197276.

**Acknowledgements** The authors would like to thank Aurélien Grolet, Olivier Thomas, Loic Salles, Alessio Colombo and Attilio Frangi for fruitful discussions related to the subject. The authors would also like to express their heartfelt gratitude to the reviewers for their invaluable contributions to the final version of this paper. Their meticulous proofreading of the manuscript and identification of minor errors in a few equations were greatly appreciated. Moreover, their constructive feedback during both stages of the review process significantly enhanced the quality of the article, both in content and presentation. We sincerely thank them and are pleased to acknowledge their insightful input and dedication.

**Competing interests** The authors declare that they have no competing interests.

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