We propose to investigate global properties of certain systems of partial differential equations of geometric importance, on spaces known as tube structures, or the product manifolds. They are constitued by systems of vector fields with symmetries that can be studied via Fourier analysis. Our aim is to determine necessary and/or sufficient conditions for their solvability, and for the regularity of their solutions, when the ambient space is a so-called Lie group, which encode extra symmetries of the equations and their solutions. Certain second-order operators associated with such systems (known as sub-Laplacians, or sums-of-squares of vector fields) will also be investigated from this point of view, since their properties are related, and also connect the former systems with applications. Both the general theory as well as concrete special cases will be studied. We will be addressing questions of fundamental importance for these systems, such as the hypoellipticity and solvability of the corresponding system of partial differential equations. The project will combine in a unique way the expertise of the Brazilian team on hypoellipticity and solvability of partial differential operators, with the expertise of the Belgian team on different aspects of the noncommutative Fourier analysis and the theory of pseudo-differential operators on Lie groups.