Real algebraic curves and differential equations

We consider the problem of the existence of branches of real algebraic curves adjacent to their isolated singular points. More precisely, the hypothetical existence of such branches in $n$-dimensional space is discussed for $n\not=2$ and $n\not=4$. The hypothesis is proved for $n=3$, and sufficient conditions for the existence of nontrivial branches that depend on the topology of the links of the corresponding algebraic manifolds are indicated. The method of proving is based on reducing these problems to the study of asymptotic trajectories of systems of differential equations of some special type. The first steps of the ergodic theory of real algebraic curves are outlined.