The aim of this paper is to take the study of Dedekind braces, that is, left braces for which every subbrace is an ideal, started in a previous paper, further. Dedekind braces A whose additive group is non-periodic are analysed. We prove sufficient conditions for A to be abelian: it is enough that every element is 2-nilpotent for the star operation; and, if A is hypermultipermutational, it suffices that the additive group of the socle is torsion-free. Both conditions can be translated in terms of set-theoretical solutions of the Yang-Baxter equation. In addition, we prove a structural theorem for the case of A to be a multipermutational brace of level 2.

All authors are supported by the grant: PID2024-159495NB-I00, funded by MICIU / AEI / 10.13039/501100011033 / FEDER, UE. The first, second, and fourth authors are supported by the grant: CIAICO/2023/007, funded by Conselleria d'Educacio, Universitats i Ocupacio, Generalitat Valenciana. The third author is very grateful to the Conselleria d'Innovacio, Universitats, Ciencia i Societat Digital of the Generalitat (Valencian Community, Spain) and the Universitat de Valencia for their financial support and grant to host researchers affected by the war in Ukraine in research centres of the Valencian Community.