We consider a three-dimensional vector field with quadratic nonlinearities and in general none of the axis plane is invariant. For our investigation, we are interesting in the case of $(1:-2:1)$ – resonance at the origin. Hence, we deal with a nine parametric family of quadratic systems and our purpose is to understand the mechanisms of local integrability. By computing some obstructions, knowing as resonant focus quantities, first we present necessary conditions that guarantee the existence of two independent local first integrals at the origin. For this reason Gröbner basis and some other algorithms are employed. Then we examine the cases where the origin is linearizable. Some techniques like existence of invariant surfaces and Jacobi multipliers, Darboux method, properties of linearizable nodes of two dimensional systems and power series arguments are used to prove the sufficiency of the obtained conditions. For a particular three-parametric subfamily, we provide conditions on the parameters to guarantee the non-existence of a polynomial first integral.

我们考虑具有二次非线性的三维向量场，并且通常没有轴平面是不变的。对于我们的调查，我们对以下情况很感兴趣$(1：-2：1)$– 原点共振。因此，我们处理二次系统的九个参数族，我们的目的是了解局部可积性的机制。通过计算一些障碍物，知道作为共振焦点量，首先我们提出了保证在原点存在两个独立的局部第一积分的必要条件。为此，采用了 Gröbner 基和一些其他算法。然后，我们检查原点可线性化的情况。利用不变曲面的存在性和雅可比乘子、Darboux 方法、二维系统可线性化节点的性质和幂级数参数等技术来证明所得条件的充分性。对于特定的三参数亚族，