This work continues our studies in properties of the mutually homogeneous functions (MHF) being a generalization of Euler homogeneous functions. MHF can be used in the synthesis of electric and magnetic fields for electron systems and ion-optical ones with special properties. A chain of functions corresponding to multiple real eigenvalues of the matrix of basic functional relations for MHF has been considered. Functional relations answering such functions were derived. General formulas for the solutions of the obtained functional relations were derived. The obtained functions were shown to be a refinement of the associated homogeneous functions introduced by Gel’fand. Typical differential and integral properties of the obtained functions were investigated, and a generalization of the Euler theorem was proved (Euler criterion) for differentiable functions.