  The Finslerian curvature is introduced in the three-dimensional isospin
space, suggesting that the isospin-pion field transforms according to
non-linear Finslerian invariance transformations. The fundamental non-linear
realization is associated with the pion field triplet. The Finslerian
Lagrangian, as well as the implied pion-field selfinteraction, involves the
first but no higher derivatives of the pion field. While the invariance
transformations are now nonlinear, the conservation laws are still meaningful
and sufficient to get the conservations for the energy-momentum and charges.
The Finsler-invariant pion-nucleon Lagrangian function is also proposed. The
Finslerian isospin-space symmetry is a pure interaction symmetry not shared by
the asymptotic fields. The full Klein-Gordon-Dirac equation is accordingly
extended. Various details of the implied Finsleroid-structure are formulated in
an explicit way.
