Integrable Hamiltonians related to the $\mathfrak{e}(3)$, $\mathfrak{so}(4)$, and $\mathfrak{so}(3,1)$ Poisson brackets

We present a brief overview of our results on new integrable cases in Hamiltonian mechanics. The paper is devoted to a description of quadratic Hamiltonians that have an additional integral of motion in the cases of linear Poisson brackets $\mathfrak{e}(3)$, $\mathfrak{so}(4)$ and $\mathfrak{so}(3,1)$.