4 problems

Differential Galois integrability conjecture. The identity component of the differential Galois group of the $k$-th order variational equations is Abelian for every…

Let $\mathrm{LVE}^m_{\phi_0}$ denote the linearized variational equation of order $m$ along the solution $\phi_0$, and let $\tilde{A}_{m+1,R}$ be the output of the reduction proced…

Fifth- and seventh-order nonintegrability conjecture. If $n$ is odd, then the fifth variational equation is not integrable. If $n$ is even and $n\geq 4$, then the seventh variation…

Nonintegrability conjecture. There exists $k\in\mathbb{N}^*$ such that the variational equation at order $k$ is not integrable.