Construction of Singular Solutions of the Lamé System in a Cone with Vanishing Condition for the External Pressure

We consider singular solutions $$\mathbf u=r^{\mu}\mathbf U(\theta,\varphi)$$ to the Lamé system of equations in an infinite circular cone with boundary condition of vanishing external loads. Such solutions are written using the Papkovich — Neuber formula in terms of three harmonic functions; representations for this functions we take in the form of linear combinations of spherical harmonics. Using the boundary condition, a set of homogeneous linear algebraic systems is obtained for the corresponding coefficients, the matrices $$\mathbb T^{(m)}$$ of which depend on parameter $$\mu$$ ; the elements of such systems are written explicitly through the Legendre functions. Thus, it is established that the sought exponents $$\mu$$ are solutions of systems of equations of the form $$\det\mathbb T^{(m)}(\mu;\theta_0)=0$$ , where the matrix has size $$2\times2$$ for $$m=0$$ and size $$3\times3$$ at $$m\geq1$$ .