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 $$\mathfrak{so}(3,1)$$ .