We study orbit-finite systems of linear equations, in the setting of sets with atoms. Our principal contribution is a decision procedure for solvability of such systems. The procedure works for every field (and even commutative ring) under mild effectiveness assumptions, and reduces a given orbit-finite system to a number of finite ones: exponentially many in general, but polynomially many when atom dimension of input systems is fixed. Towards obtaining the procedure we push further the theory of vector spaces generated by orbit-finite sets, and show that each such vector space admits an orbit-finite basis. This fundamental property is a key tool in our development, but should be also of wider interest.

翻译：我们研究线性方程的轨道-无限系统,用原子来设定数组。我们的主要贡献是这类系统可溶性的决策程序。该程序在微小的有效性假设下适用于每个领域(甚至移动环),并将特定轨道-无限系统减少到若干有限的领域:一般数量极多,但当输入系统的原子维度固定下来时,则多为多为多为。为了获得程序,我们进一步推动轨道-无限空间产生的矢量空间理论,并表明每个矢量空间都接受一个轨道-无限基础。这一基本属性是我们发展的关键工具,但也应该具有更广泛的意义。