by **Stepan Orevkov**

**Abstract**

In the paper [S.Yu.Orevkov, *An example in connection to the Jacobian
Conjecture,* Math. Notes, **47**(1990) no. 1-2, 82-88], we
constructed an example of an open complex surface *U,* a compact
rational curve *L* on *U * of the self-intersection
*L.L*=+1 and an immersion $f:U\to C^2$ defined by
two functions which are holomorphic on *U-L* and meromorphic
on *U*, such that f is not an embedding (if *U *
were isomorphic to a domain in $C^2$ such an immersion would
provide automatically a counter-example to the well-known Jacobian Conjecture).
In this paper we construct a similar example which satisfies the additional
property that the restriction of * f* to the boundary of
*U* is an immersion of the 3-sphere into $C^2$ which
is regularly homotipic to the standard embedding. We prove that the
constructed immersion can not be extended to a counter-example to the Jacobian
Conjecture, analysing coefficients of polynomials *P*(*x,y*),
*Q*(*x,y*) with constant jacobian which might realize such
an immersion. As a corollary, we obtain an example of a pseudo-convex
immersion of a 3-sphere into the complex plane which is regularly homotopic
to the standard embedding but which is not homotopic through pseudo-convex
immersions.