This work completes the study, initiated by the author in [part I, Commun. Nonlinear Sci. Numer. Simul. 16, No. 8, 3015–3023 (2010; Zbl 1235.34108)], of symmetry algebras (of Lie point symmetries) for systems \[ y''=M_1y'+M_2{ y}+M_3(t), \] where \( y, M_3(t)\in\mathbb R^n\), the \(\prime\) denotes the derivative with respect to \(t\), and \(M_1,M_2\) are \(n\times n\) real matrices. Through appropriate changes of variables, this problem is reduced to the investigation of symmetry algebras of systems \[ x'' =J{ x}, \] where \(J\) is an \(n\times n\) matrix in Jordan form and \({ x}\in\mathbb R^n\). In this work, the author completes previous results by studying the case for nondiagonal \(J\) and \(n > 4\). Special attention is paid to the case \(n=5\).
As a first result, the author proves that for a Jordan matrix \(J_\lambda\) with a unique real eigenvalue \(\lambda,\) the dimension of the symmetry algebra \(\mathcal{L}_\lambda\) of the system becomes \[ \dim\mathcal{L}_\lambda=(2+p_0)n-\!\!\sum_{i=1}^{p_0-1}\sum_{k=j+1}^{p_0}(m_j-m_k)+\epsilon_\lambda, \] where \(\epsilon_\lambda=2\) when \(\lambda=0\) and \(\epsilon_\lambda=1\) otherwise, \(p_0\) is the number of Jordan blocks \(J^k\), with \(k=1,\ldots,p_0\), of \(J_\lambda\), and \(J^k\) has dimension \((m_k+1)\times (m_k+1)\). Several related results are provided. Next, the paper addresses the study of symmetry algebras \(\mathcal{L}\) of systems \(x'' =J{ x}\), where \(J\) has two complex conjugated eigenvalues.
Using previously obtained results, the author calculates the dimension of the Lie algebra of symmetries of the system associated to any \(J\). Moreover, the Levi factors of such Lie algebras are also investigated and the solvability of \(\mathcal{L}\) is determined from the form of \(J\).