There are solvable linear complementarity problems which can not be solved by the algorithm of Lemke. In such a case this algorithm injures at an edge-ray. To continue the search we permit also passages over edge-rays of the feasible region. These can be executed by the simplex-method. It is possible to keep Lemke's combinatoric pivoting rule. The extended algorithm solves all problems solvable also by Lemke' method and others.