  The main result is the identification of the orthogonal complement of the
subalgebra of conformal vector field inside the algebra of all vector fields of
a compact flat 2-manifold. As a fundamental tool, the complete Hodge
decomposition for manifold with boundary is used. The identification allows the
derivation of governing differential equations for variational problems on the
space of conformal vector fields. Several examples are given. In addition, the
paper also gives a review, in full detail, of already known vector field
decompositions involving subalgebras of volume preserving and symplectic vector
fields.
