Techniques of invariant theory such as Molien generating functions and integrity bases are more efficient mathematical tools than those of basic group theory based on projectors for the construction of symmetry-adapted polynomials in the symmetrized coordinates of a molecular system. The present article extends our previous work to the construction of polynomial bases that span a non--totally symmetric irreducible representation. Electric or magnetic observables can carry such irreducible representations, a common example is given by the electric dipole moment surface. The elementary generating functions and their corresponding integrity bases, where both the initial and the final representations are irreducible, are the building blocks of the fast algorithm presented in this article. The generating functions for the full initial representation of interest are built recursively from the elementary generating functions. Integrity bases which can be used to generate in the most economical way symmetry-adapted polynomial bases, are constructed alongside in the same fashion. The method is illustrated in detail on XY4 type of molecules. Explicit integrity bases for all five possible final irreducible representations of the tetrahedral group have been calculated and are given in the Supporting Information to this paper.