Horospherical splittings of $\mathfrak{g}$ and related Poisson commutative subalgebras of $\mathcal{S}(\mathfrak{g})$

Let a Lie algebra $\mathfrak{q}$ be a linear sum of two complementary subalgebras $\mathfrak{h}$ and $\mathfrak{r}$. We continue our investigations initiated in (J. London Math. Soc. 103 (2021), 1577-1595), where compatible Poisson brackets associated with splitting $\mathfrak{q}=\mathfrak{h}\oplus \mathfrak{r}$ and related Poisson-commutative subalgebras of the symmetric algebra $\mathcal{S}(\mathfrak{q})$ are studied. In this article, we further develop the general theory and study in more details splittings of the reductive Lie algebras such that both $\mathfrak{h}$ and $\mathfrak{r}$ are solvable horospherical subalgebras. We also derive some results of the Adler-Kostant-Symes theory using our approach.