The version of record of this article, first published in Journal of Marine Science and Technology (2025), is available online at Publisher’s website: https://doi.org/10.1007/s00773-025-01081-z. The International Maritime Organization developed the Second Generation Intact Stability Criteria (SGISC) with the objective of mitigating maritime incidents attributable to roll motion. Given that surf-riding precedes broaching, the SGISC uses vulnerability criteria for broaching based on surf-riding dynamics. The surf-riding threshold is derived by Melnikov’s method in the level-2 vulnerability criterion. In this paper, the authors discuss the uniqueness of the approximate analytical solutions to be used as the surf-riding threshold in the SGISC, based on the relationship between Melnikov’s analysis and its physical interpretation.