Summary: We study the problem of the flutter of a conical clamped-free shell under the action of a supersonic gas flowing out from it. The shell is assumed to be elastic, homogeneous, and isotropic, and is described by equations of the technical theory of shells in mixed form. In such a description, one must find two scalar functions on the middle surface: the normal deflection and the function of membrane forces. Therefore, boundary conditions must be formulated in terms of these functions and their derivatives. It is extremely important that, in the absence of interaction with the flow, the boundary conditions ensure that the mechanical system is conservative.