  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 hyperelliptic integrals of all three kinds. The result of the
inversion is defined locally, 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, three
and four are considered in detail. The method is exemplified by the particle
motion associated with genus one elliptic and genus three hyperelliptic curves.
Applications are for instance solutions to the geodesic equations in the
space-times of static, spherically symmetric Ho\v{r}ava-Lifshitz black holes.
