K-meson production and isospin-symmetry in $pp$ and $AA$ collisions

The production of $K$ mesons in $pp$ and $AA$ collisions at high energies is analyzed. We have showed that the ratio $R_K~=~\frac{\sigma_{K^+}+\sigma_{K^-}}{2\sigma_{K^0_S}}$ calculated within the similarity approach, including the gluon TMD at low QCD scales\cite{LMZ:2024}, and using the Lund model in the form of MC generator PYTHIA \cite{Lund} is above 1 even at the conservation of the isospin-symmetry. It is about 1.18 at $\sqrt{s}=$11.9 GeV (the NA61/SHINE data \cite{Isotp_vial}) and it decreases up to 1, when the energy increases up to a few TeV (ALICE data). The reason of this is related to the dynamics of the kaon production. This energy behavior of $R_K$ practically does not depend on the sort of the beam or target. \begin{thebibliography}{99} \bibitem{LMZ:2024} G.I.~lykasov, A.I.~Malakhov, A.A.~Zaitsev, Eur.Phys. J. A{\bf 60}, 239 (2024). \bibitem{Lund} T.Sjoestrand, Comp. Phys. Commun, {\bf 82}, 4 (1994). \bibitem{Isotp_vial} The NA61/SHINE Collaboration, F.~Giacosa, M.~ Gorenstein,R.~Poberezhniuk, S.~Samanta, Nature Communications, {\bf 16}, 2849 (2025). \end{thebibliography}rXiv:2312.06572v6 [nucl-ex].