The concept of scalar hyperplane-dependent fields is defined on the basis of the author's earlier treatments of the position operators for relativistic particles. Such fields are a special case of fields over homogeneous spaces of the PonincarB group and are related to infinite families of integer spin particles with spin-dependent mass spectra. The intrinsic parity of the particles and degeneracy in the mass spectrum is examined. The Lagrangian formalism, including Noether's theorem, is developed and explored. I t is shown that the existence of a Lagrangian formalism yields a normalization condition that is incompatible with the existence of spacelike solutions. All is in preparation for the sequel to this paper which considers a specific model with Bose statistics and nondegenerate mass spectrum.