  In this paper, we examine the structure of systems that are weighted
homogeneous for several systems of weights, and how it impacts the computation
of Gr\"obner bases. We present several linear algebra algorithms for computing
Gr\"obner bases for systems with this structure, either directly or by reducing
to existing structures. We also present suitable optimization techniques.
  As an opening towards complexity studies, we discuss potential definitions of
regularity and prove that they are generic if non-empty. Finally, we present
experimental data from a prototype implementation of the algorithms in
SageMath.
