It is generally accepted that the only type of motion present in a symmetric Euler gyroscope (SEG) is regular precession. This paper proves that regular precession is not the only type of motion present, but corresponds only to the well-known initial coordinated Euler angles. At any other initial angles, motions that differ from regular precession occur. In the article, the problem is solved analytically in two stages: first, angular velocities of the gyroscope are determined using differential dynamic equations, at the second stage, as a result of integration of differential matrix kinematic and differential matrix Poisson equations (both with periodic coefficients), final relations about the SEG motion with arbitrary initial Euler angles are derived. Periodic coefficients are the SEG angular velocities that are found as a solution to the dynamic equations. From the obtained general formulas, special formulas of regular precession for particular coordinated initial Euler angles that coincide with the well-known ones are derived.