The talk deals with the Neumann problem for elliptic systems of second order differential equations in a polyhedral cone. In particular, the Green matrix for this problem is studied. One of the main results are point estimates for the Green matrix and its derivatives. These estimates are used to prove

1. the solvability of the Neumann problem in weighted $L_p$ Sobolev and H\"older spaces,
2. the existence of weak solutions in weighted $L_p$ Sobolev spaces,
3. regularity assertions for the solutions.

As an example, the Neumann problem for the Lam\'e system is considered. A feature of this problem is that $\lambda=1$ is always an eigenvalue of the operator pencils characterizing the singularities of the solutions near the edges. Therefore, an $H^2$ regularity result for the solutions is not obvious.