  We introduce families of quasi-rectifiable vector fields and study their
geometric and algebraic aspects. Then, we analyse their applications to systems
of partial differential equations. Our results explain, in a simpler manner,
previous findings about hydrodynamic-type equations. Facts concerning families
of quasi-rectifiable vector fields, their relation to Hamiltonian systems, and
practical procedures for studying such families are developed. We introduce and
analyse quasi-rectifiable Lie algebras, which are motivated by geometric and
practical reasons. We classify different types of quasi-rectifiable Lie
algebras, e.g. indecomposable ones up to dimension five. New methods for
solving systems of hydrodynamic-type equations are established to illustrate
our results. In particular, we study hydrodynamic-type systems admitting
$k$-wave solutions through quasi-rectifiable Lie algebras of vector fields. We
develop techniques for obtaining the submanifolds related to quasi-rectifiable
Lie algebras of vector fields and systems of partial differential equations
admitting a nonlinear superposition rule: the PDE Lie systems.
