Gr\"obner-Shirshov bases of Rota-Baxter algebra of weight $\lambda$ with spectrum lying in $\{0,-\lambda\}$

It is known that if $A$ is a finite-dimensional unital algebra equipped with a Rota-Baxter operator $R$ of weight $\lambda$, then spectrum of $R$ is a subset of $\{0,-\lambda\}$. We are interested on finding all consequences of the Rota-Baxter relation and the relation of the form $R^k(R+\lambda{\rm id})^l = 0$. In 2024, H.~Qiu, S. Zheng, Y. Dan solved this problem for $k = l = 1$ and $\lambda\neq0$. We find a Gr\"obner-Shirshov basis of the ideal generated by these two relations in general case.