Yesrerday immediate after the first version of your question I commented on your question but unfortunately I can not find my comments. any way my comment was identical to my existing answer

Thank you a lot for the references

Dear Prof. Bryant Thank you very much for your very interesting answer. It would be kind of you if you expand your answer and introduce the references.

@YCor But your answer in the case of G- conjugacy would be also very welcome and appreciated

@YCor No I mean it is conjugate in a wider class $C^{\infty}$ diffeomorphisms of $\mathbb{R}^2$. In fact the motivation comes from the linked post so the conjugacy is meant $C^\infty$ dynamical conjugate

@YCor Thank you for your attention. the reason for the caution of the last part of the question is that it is possible that we would have an effective action of $H\times K$ or a semidirectt product $H\rtimes K$ on the plane (for a Lie group $K$). So this would result to a possible trivial answer to the first part. But now I realize that you think the first part is not an immediate question