Substantial condition for the fourth first integral of the rigid body problem

The aim of this article is to study the possibility of obtaining the fourth integral for the motion of a rigid body about a fixed point in the presence of a gyrostatic moment vector. This problem is governed by a system consisting of six nonlinear differential equations from first order, as well as three first integrals. A most important condition for a function F, depending on all the body variables, to be that integral is presented. This work can be considered a mainstreaming of previous works. The importance of this work lies in several applications of the rigid body problem and gyroscopic motion in different areas, such as physics, engineering and industrial applications, for example, in aircraft specially designed to use the auto-pilot function, calculating aircraft turns about various axes of operation (pitch, yaw and roll), and maintaining aircraft orientation.