I hope I've got the meaning of your question correctly: you are interested in non-Hamiltonian systems where the energy is nevertheless conserved. If this is the case, then a fairly broad class of such examples is provided by the so-called quasi-potential systems which are a rather natural generalization of Hamiltonian ones; see e.g. http://www.mai.liu.se/~halun/research/cofactor_systems/ for an introduction.

One extensively studied class of quasi-potential systems is formed by the so-called cofactor systems; see e.g. the Ph.D. thesis by Hans Lundmark (http://www.math.liu.se/~halun/papers/hlthesis.pdf ) as well as e.g. a fairly recent paper Non-Hamiltonian systems separable by Hamilton-Jacobi method by Maciej Blaszak and Krzysztof Marciniak (http://www.researchgate.net/publication/222827760_Non-Hamiltonian_systems_separable_by_HamiltonJacobi_method or http://webstaff.itn.liu.se/~krzma/publications/bicofactorJGeomPhys.pdf for the published version) and references therein for further details.

Here is a simple explicit example (see PDF page 18 of Lundmark's thesis):

q'=p, r'=s, p'= a q -r/q^5, s'=4 a r-b, (*)

where the prime indicates the time derivative and a,b are parameters. This system is not canonically Hamiltonian but it has an energy integral (in addition to another integral quadratic in the momenta p and s):

E=− r p^2 + q ps − a q^2 r+r^2/(2q^4)+(b/2) q^2.

Note however (see PDF page 19 of the same thesis) that the system (*) admits a (non-canonical !) Hamiltonian representation in an extended phase space.

Article B̷laszak M, Non-Hamiltonian systems separable by Hamilton-Ja...