  This work represents a PhD thesis concerning three main topics. The first one
deals with the study and applications of Lie systems with compatible geometric
structures, e.g. symplectic, Poisson, Dirac, Jacobi, among others. Many new Lie
systems admitting Vessiot--Guldberg Lie algebras of Hamiltonian vector fields
relative to the above mentioned geometric structures are analyzed and their
importance is illustrated by their appearances in physical, biological and
mathematical models. The second part details the study of Lie symmetries and
reductions of relevant hierarchies of differential equations and their
corresponding Lax pairs. For example, the Cammasa-Holm and Qiao hierarchies in
2+1 dimensions. The third and last part is dedicated to the study of reciprocal
transformations and their application in differential equations appearing in
mathematical physics, e.g. Qiao and Camassa-Holm equations equations appearing
in the second part.
