We prove an existence theorem for the statistical elasticity theory equation for a homogeneous incompressible medium and its extension to the second and third boundary value problem case. We demonstrate, in the case of the first, second, and third problems that, as $\lambda\to\infty$ the solution of the elasticity theory equation with Lam'e constants $\lambda$ and $\mu$ converges to the solutions of the respective equations for incompressible material. An existence theorem in the rectangle is demonstrated for the third boundary value problem in $W^2_q$ .