Counting isotropic tangent lines of hypersurfaces

Consider the standard symplectic (R2n,ω0), a point p∈R2n and an immersed closed orientable hypersurface Σ⊂R2n∖{p}, all in general position. We study the following passage/tangency question: how many lines in R2n pass through p and tangent to Σ parallel to the 1-dimensional characteristic distribution ker⁡(ω0|TΣ)⊂TΣ of ω0. We count each such line with a certain sign, and present an explicit formula for their algebraic number. This number is invariant under regular homotopies in the class of a general position of the pair (p,Σ), but jumps (in a well-controlled way) when during a homotopy we pass a certain singular discriminant. It provides a low bound to the actual number of these isotropic lines.