Integrable systems related to deformed \(\mathfrak{so}(5)\) (Q401232)

A pencil of Lie brackets on a vector space \({\mathfrak{g}}\) defines compatible Lie-Poisson structures on the dual space \(\mathfrak{g}^{*}\) as well as many examples of integrable Hamiltonian systems on \(\mathfrak{g}^{*}\). The authors consider a two-parametric family of Lie algebras \(\mathfrak{g}=\text{so}_{\lambda,\alpha}(5)\) of \((5 \times 5)\)-matrices NEWLINE\[NEWLINE\begin{pmatrix} 0 & \alpha b &\alpha\lambda\vec{u}\,^T \\ -b & 0 & \lambda\vec{w}\,^T \\-\vec{u} & -\vec{w} & \delta \end{pmatrix} \quad \text{where} \;\lambda, \alpha, b \in \mathbb{R},\lambda\alpha\not= 0; \;\vec{u},\vec{w} \in \mathbb{R}^3; \;\delta \in \text{so}(3).NEWLINE\]NEWLINE The case \(\lambda>0,\alpha>0\) corresponds to the special orthogonal algebra \(\text{so}(5)\), the case \(\lambda<0,\alpha<0\) corresponds to the anti-de Sitter algebra \(\text{so}(3,2)\), and \(\lambda<0>\alpha\) corresponds to de Sitter algebra \(\text{so}(1,4)\). The dual algebra \(\text{so}_{\lambda,\alpha}^*(5)\) can be identified with the upper-triangular \((5 \times 5)\)-matrices.NEWLINENEWLINEThe authors investigate the associated integrable Hamiltonian NEWLINENEWLINE\[NEWLINE\begin{aligned} h &= \gamma \big[ \alpha(\alpha\lambda^2 q_{-1}^2 +\lambda^2 q_0^2 +\varepsilon\vec{q}\,^2) \vec{p}\,^2 +(p_{-1}^2+\alpha p_0^2) (\vec{q}\,^2 + \varepsilon q_0^2 + \alpha\varepsilon q_{-1}^2) \\ & -\alpha\varepsilon(q_{-1}p_{-1}+q_0 p_0 +\vec{p}\cdot\vec{q}\,)^2 -2\alpha (\lambda-\varepsilon) (q_{-1}p_{-1}+q_0 p_0) (\vec{p} \cdot \vec{q}\,) \big] \\& + \nu (\lambda-\varepsilon)^2 (\alpha q_{-1}p_0-q_0 p_{-1})^2 (\vec{p} \times \vec{q}\,)^2 \end{aligned}NEWLINE\]NEWLINE where \(\lambda,\alpha,\gamma,\varepsilon,\nu\) are fixed parameters, \(\vec{p}=(p_1,p_2,p_3)^T\), \(\vec{q}=(q_1,q_2,q_3)^T\) and \(p_s=p_s(t)\), \(q_s=q_s(t)\), \(s=-1,\ldots,3\) are the Hamiltonian variables. It turns out that \(p_{-1},p_0,q_{-1},q_0\) satisfy an homogeneous first order system of four ODE's with constant elements and can be integrated immediately in terms of \(\cos\), \(\sin\), \(\cosh\), \(\sinh\) with arguments linear on the time \(t\). Then \(\vec{p},\vec{q}\) satisfy an analogous but already non-autonomous system of six ODE's. To solve explicitly this system it should be found an explicit intersection of three quadrics and a quartic in \(\mathbb{R}^4\).