  The description of many dynamical problems like the particle motion in higher
dimensional spherically and axially symmetric space-times is reduced to the
inversion of a holomorphic hyperelliptic integral. The result of the inversion
is defined only locally, and is done using the algebro-geometric techniques of
the standard Jacobi inversion problem and the foregoing restriction to the
$\theta$--divisor. For a representation of the hyperelliptic functions the
Klein--Weierstra{\ss} multivariable sigma function is introduced. It is shown
that all parameters needed for the calculations like period matrices and
Abelian images of branch points can be expressed in terms of the periods of
holomorphic differentials and theta-constants. The cases of genus two and three
are considered in detail. The method is exemplified by particle motion
associated with a genus three hyperelliptic curve.
