In this paper, we show that to find the traveling wave solutions for the Krichever–Novikov equation, we only need to consider a spatial form F-VI of the fourth-order differential equations in the polynomial class having the Painlev´e property given by [Cosgrove, 2000]. By using the method of dynamical systems to analyze the dynamical behavior of the traveling wave solutions in some two-dimensional invariant manifolds, various exact solutions such as solitary wave solution, periodic wave solutions, quasi-periodic wave solutions and uncountably infinitely many unbounded wave solutions are obtained.