  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.
