  This article studies the global hypoellipticity of a class of overdetermined
systems of pseudo-differential operators defined on the torus. The main goal
consists in establishing connections between the global hypoellipticity of the
system and the global hypoellipticity of its normal form. It is proved that an
obstruction of number-theoretical nature appears as a necessary condition to
the global hypoellipticity. Conversely, the sufficiency is approached
ana\-lyzing three types of hypotheses: a H\"{o}rmander condition, logarithmic
growth and super-logarithmic growth.
