Complete integrability versus symmetry (Q2872411)

A well known theorem on integration of differential equations in a domain \(\Omega\subset \mathbb R^n\) due to Lie says that if one have \(n\) linearly independent vector fields \(X_1, \dots, X_n\) in \(\Omega\) that generates a solvable Lie algebra under commutation: \([X_1, X_j ] = c^1_{1,j} X1\), \([X2, X_j] = c^1_{1, j} X_1 + c^2_{2, j} X_2\), \(\dots\), \([X_n, X_j ] = c^1{1, j} X_1 + c^2_{2, j} X_2 +\dots+c^n_{n, j} X_n\), then the differential equation \(\dot x = X_1(x)\) is solvable by quadratures. In fact, each of the differential equations \(\dot x = X_j (x)\) is integrable by quadratures [\textit{V. V. Kozlov}, Differ. Equ. 41, No. 4, 588--590 (2005); translation from Differ. Uravn. 41, No. 4, 553--555 (2005; Zbl 1095.34501)]. Solving the system \(\dot x=X_1(x)\) by quadratures implies the existence (at least locally) of \(n - 1\) functionally independent first integrals of the system. Also, the solvable Lie algebra generated by \(X_1,\dots,X_n\) integrates to a solvable \(n\)-dimensional Lie group that maps the trajectories of the system to the trajectories of the system.NEWLINENEWLINEIn this paper, an interesting converse problem is considered. Assume that the system \(\dot x=X(x)\) has \(n-1\) independent first integrals in \(\Omega\). The main result states that under some reasonable assumptions one can construct a vector field \(\tilde X\) defined on an open dense set \(\Omega_{0}\subset\Omega\), such that to each vector field \(\bar Y\), with \([\bar Y,\tilde X]=0\), there exists a real scalar function \(\mu =\mu(\bar Y)\) and a vector field \(Y = Y (\bar Y)\) such that \([X, Y] = \mu X\). Therefore, there exists a one-parameter Lie group which permutes the trajectories of the system \(x = X(x)\). This result is a generalization of a similar one for planar vector fields [\textit{J. Giné} and \textit{J. Llibre}, Z. Angew. Math. Phys. 62, No. 4, 567--574 (2011; Zbl 1263.34046)]. The proof is based on a local normal form for Nambu-Hamiltonian dynamical systems given by the author in [J. Geom. Phys. 62, No. 5, 1167--1174 (2012; Zbl 1238.37016)].