Polynomial vector fields \(X:\mathbb {R}^3\rightarrow \mathbb {R}^3\) that have invariant algebraic surfaces of the form

are considered. We prove that trajectories of X on M are solutions of a constrained differential system having \(\mathcal {I}=\{f(x,y)=0\}\) as impasse curve. The main goal of the paper is to study the flow on M near points that are projected on typical impasse singularities. The Falkner–Skan equation (Llibre and Valls in Comput Fluids 86:71–76, 2013), the Lorenz system (Llibre and Zhang in J Math Phys 43:1622–1645, 2002) and the Chen system (Lu and Zhang in Int J Bifurc Chaos 17–8:2739–2748, 2007) are some of the well-known polynomial systems that fit our hypotheses.

具有形式不变的代数曲面的多项式向量字段\（X：\ mathbb {R} ^ 3 \ rightarrow \ mathbb {R} ^ 3 \）

被考虑。我们证明X在M上的轨迹是具有\（\ mathcal {I} = \ {f（x，y）= 0 \} \）作为僵局曲线的约束微分系统的解。本文的主要目的是研究在典型僵局奇异点上投影的M个近点上的流动。Falkner–Skan方程（计算机流体中的Llibre和Valls，2013年：86：71–76），Lorenz系统（Llibre和Zhang，J Math Phys 43：1622–1645，2002）和Chen系统（Lu和Zhang，Int J Bifurc Chaos 17–8：2739–2748，2007）是一些符合我们假设的著名多项式系统。