  As is well known, energy is generally deemed as one of the most important
physical invariants in many conservative problems and hence it is of remarkable
interest to consider numerical methods which are able to preserve it. In this
paper, we are concerned with the energy-preserving integration of non-canonical
Hamiltonian systems by continuous-stage methods. Algebraic conditions in terms
of the Butcher coefficients for ensuring the energy preservation, symmetry and
quadratic-Casimir preservation respectively are presented. With the presented
condition and in use of orthogonal expansion techniques, the construction of
energy-preserving integrators is examined. A new class of energy-preserving
integrators which is symmetric and of order $2m$ is constructed. Some numerical
results are reported to verify our theoretical analysis and show the
effectiveness of our new methods.
