Analytic integrability around a nilpotent singularity (Q1737543)

A polynomial \(f(x,y)\) is called quasi-homogeneous of type \(\mathbf{t}=(t_1,t_2)\in \mathbb{N}^2\) and degree \(k\) if \(f(\alpha^{t_1} x, \alpha^{t_2} y )=\alpha^k f (x,y)\) for arbitrary \(\alpha>0\). Denote by \(\mathcal{P}_k^{\mathbf{t}}\) the vector space of quasi-homogeneous polynomials of type \(\mathbf{t}\) and degree \(k\) and by \(\mathcal{Q}_k^{\mathbf{t}}\) the vector space of polynomial quasi-homogeneous vector fields of type \(\mathbf{t}\) and degree \(k\). Note that any \( \mathbf{F}_j\in \mathcal{Q}_k^{\mathbf{t}}\) can be uniquely written as \[ \mathbf{F}_j =\mathbf{X}_{h_j} + \mu_j\mathbf{D}_0, \] where \(\mu_ j =\frac 1{j+|\mathbf{t}|} \operatorname{div}\mathbf{F_j}\in \mathcal{P}_j^{\mathbf{t}}\), \(h_j = \frac{1}{j+|t|} \mathbf{D}_0 \wedge \mathbf{F}_j \in \mathcal{P}_j^{\mathbf{t}}\), \(\mathbf{D}_0 = (t_1 x, t_2y)^T\) , and \(\mathbf{X}_{h_j}=(-\partial h_j/\partial y, \partial h_j/\partial x)^T\) is the Hamiltonian vector field with Hamiltonian function \(h_j\). Let \(\mathbf{F} = \sum_{j\ge r} \mathbf{F}_j\), \( \mathbf{F}_j\in\mathcal{Q}_k^{\mathbf{t}}\) be an analytic nilpotent vector field such that the origin of \(\dot x = \mathbf{F}_r (x)\) is isolated and \(\mathbf{F}_r\) is polynomially integrable. The main results of the paper are the following theorems. Theorem 1. \(\mathbf{F}\) is analytically integrable if and only if it is formally orbitally equivalent to \(\mathbf{F}_r\). Theorem 2. \(\mathbf{F}\) is analytically integrable if and only if there exist a formal vector field \(\mathbf{G} =\sum_{j\ge 0}\mathbf{G}_j \), \( \mathbf{G}_j\in \mathcal{Q}_k^{\mathbf{t}}\), \(\mathbf{G}_0 =\mathbf{D}_0\) and a formal scalar function \(\mu\), \(\mu(\mathbf{0}) = r\) such that \([F, G] = \mu F\) (hence \(\mathbf{F}\) has a Lie symmetry).