  This article is devoted to finding classical point-particle equivalents for
the fermion sector of the nonminimal Standard-Model Extension (SME). For a
series of nonminimal operators, such Lagrangians are derived at first order in
Lorentz violation using the algebraic concept of Gr\"obner bases. Subsequently,
the Lagrangians serve as a basis for reanalyzing the results of certain
kinematic tests of Special Relativity that were carried out in the last
century. Thereby, a number of new constraints on coefficients of the nonminimal
SME is obtained. In the last part of the paper we point out connections to
Finsler geometry.
