Using the method of discontinuous solutions, the problem of the stressed state of an elastic cone, weakened by cracks, is reduced to a system of one-dimensional integro-differential equations, specified on parts of the conical surfaces where the cracks are situated. There can be an arbitrary number of such surfaces and parts. The proposed scheme is realized using the example of the problem of the torsion of a cone, weakened by a semi-infinite conical crack, subjected to the action of an arbitrary load (including the application of a centre of rotation at the cone apex. An exact solution of this problem is obtained and a formula is given for the stress intensity factor. Since there is no solution in the literature of the problem of the stressed state of a cone without cracks due to the action of a centre of rotation, a solution is also given of this problem using the new integral transformation obtained here. It can also be used to solve problems of the stressed state of cones truncated along spherical surfaces. It follows from the problem of the stressed state of a cone, loaded with centre of rotation at the apex, which is solved here, that with type of loading the stress is everywhere equal to zero inside the cone, and hence a conical crack does not weaken the cone.