Navier equations are used to describe the deformation of a homogeneous, isotropic and linear elastic medium in the absence of body forces. Mathematically, the system is a natural vector (field) $O(n,\mbb{ R})$-invariant generalization of the classical Laplace equation, which physically describes the vibration of a string. In this paper, we decompose the space of polynomial solutions of Navier equations into a direct sum of irreducible $O(n,\mbb{ R})$-submodules and construct an explicit basis for each irreducible summand. Moreover, we explicitly solve the initial value problems for Navier equations and their wave-type extension--Lam\'e equations by Fourier expansion and Xu's method of solving flag partial differential equations.
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