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.