We consider the motion about the center of mass of a solid body subjected to a newtonian field of forces and we seek those solutions of the Euler-Poisson equations, governing the motion, which make stationary the integral of the energy subordinate!y to the existence of the other known integrals. So we obtain a solution in which the cosine of the angle of mutation is furnished in function of the time by means of an elliptic integral. This solution contains four arbitrary constants and therefore it defines a class of $\infty^{4}$ possible motions for the considered solid body.