Integrable Hamiltonian systems with incomplete flows and Newton's polygons

We study the Hamiltonian vector field $v=(-\partial f/\partial w,\partial f/\partial z)$on$\mathbb C^2$,where$f=f(z,w)$ is a polynomial in two complex variables, which is non-degenerate with respect to its Newton’s polygon. We introduce coordinates in four-dimensional neighbourhoods of the “points at infinity”, in which the function $f(z,w)$ and the 2-form $dz\wedge dw$ have a canonical form. A compactification of a four-dimensional neighbourhood of the non-singular level set $T_0=f^{-1}(0)$ of $f$ is constructed. The singularity types of the vector field $v{\mathrm{\mid }}_{T_0}$ at the “points at infinity” in terms of Newton’s polygon are determined.