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

Inversion of hyperelliptic integrals of arbitrary genus with application to particle motion in General Relativity