## Complete integrability versus symmetry.(English)Zbl 1287.37039

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 [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.
In 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 [J. Giné and 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)].

### MSC:

 37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests 34A05 Explicit solutions, first integrals of ordinary differential equations 34C14 Symmetries, invariants of ordinary differential equations

### Citations:

Zbl 1095.34501; Zbl 1263.34046; Zbl 1238.37016
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### References:

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