Consider a smooth 2-dimensional surface $S$ with coordinates $(t,x)$ and pseudo-Riemann metric, i.e., the quadratic form
\begin{equation*}
Q(dt,dx) = a(t,x) dx^2 + 2b(t,x) dx dt + c(t,x) dt^2
\end{equation*}
with smooth ($C^{\infty}$) coefficients. The discriminant of $Q$ is $\Delta = b^2 - ac$, and in generic case the equation $\Delta=0$ defines a smooth curve $D$ on $S$. The form $Q$ is degenerated (isotropic) at the points of $D$, they are called parabolic points. The curve $D$ splits the surface $S$ into two parts, where the signature of $Q$ is constant. In elliptic domain $E: \Delta 0$) or negative ($a 0$ the form $Q$ is indefinite, and the tangent space $T_q S$ at each point $q \in H$ contains the isotropic cone $V_q$, which consists of two different lines. The isotropic cone $V_q$ splits the space $T_q S$ into two parts: timelike and spacelike directions.

Consider geodesics on $S$ induced by the pseudo-Riemann metric $Q$. They are the extremals of the length (with respect to $Q$) functional, i.e., the extremals of the Euler-Lagrange equation with Lagrangian $L = \sqrt{F}$, where
$$
F(t,x,p) = a(t,x) p^2 + 2b(t,x) p + c(t,x), \quad p = \frac{dx}{dt}.
$$
The parabolic points of $S$ appear the singular points of this equation. In this talk we'll consider geodesics in the neighborhood of a typical parabolic point, where the form $Q$ is degenerated generically.