The non-commutative integrability (NCI) is a property fulfilled by some Hamiltonian systems that ensures, among other things, the exact solvability of their corresponding equations of motion. The latter means that an “explicit formula” for the trajectories of these systems can be constructed. Such a construction rests mainly on the so-called Lie theorem on integrability by quadratures. It is worth mentioning that, in the context of Hamiltonian systems, the NCI has been for around 40 years, essentially, the unique criterium for exact solvability expressed in the terms of first integrals (containing the usual Liouville–Arnold integrability criterium as a particular case). Concretely, a Hamiltonian system with n degrees of freedom is said to be non-commutative integrable if we know a set of independent first integrals \(F_{1},\ldots ,F_{l}\) such that: the kernel of the \(l\times l\) matrix with coefficients \(\left\{ F_{i},F_{j}\right\} \), where \(\left\{ \cdot ,\cdot \right\} \) denotes the canonical Poisson bracket, has dimension \(2n-l\) (isotropy); and each bracket \(\left\{ F_{i},F_{j}\right\} \) is functionally dependent on \(F_{1},\ldots ,F_{l}\) (closure). In this paper, we develop two procedures for constructing the trajectories of a Hamiltonian system which only require isotropic first integrals (closure condition is not needed). One of them is based on an extended version of the geometric Hamilton–Jacobi theory, and does not rely on the above mentioned Lie’s theorem. We do all that in the language of functions of several variables.

非交换可积性（NCI）是某些汉密尔顿系统满足的一个特性，除其他事项外，这些汉密尔顿系统可确保其相应运动方程的精确可解性。后者意味着可以为这些系统的轨迹构造一个“显式”。这样的构造主要基于所谓的关于积分的可积性的李定理。值得一提的是，在哈密顿系统的背景下，NCI实质上已经存在了约40年，其精确可溶性的唯一标准是以第一积分表示的（在特殊情况下，包含通常的Liouville-Arnold可积性标准）。具体而言，具有n的哈密顿系统如果我们知道一组独立的第一积分\（F_ {1}，\ ldots，F_ {l} \）这样，则自由度是非可交换可积的：\（l \ times l \ ）系数为\（\ left \ {F_ {i}，F_ {j} \ right \} \ ）的矩阵，其中\（\ left \ {\ cdot，\ cdot \ right \} \）表示规范的Poisson括号，尺寸为\（2n-l \）（各向同性）; 并且每个括号\（\ left \ {F_ {i}，F_ {j} \ right \} \）在功能上都取决于\（F_ {1}，\ ldots，F_ {l} \）（闭包）。在本文中，我们开发了两个过程来构造哈密顿系统的轨迹，这些过程仅需要各向同性的第一积分（不需要闭合条件）。其中之一是基于几何汉密尔顿-雅各比理论的扩展版本，并且不依赖于上述李定理。我们用几个变量的函数语言来完成所有这些工作。