The transformation of a set of arbitrary ordinary differential equations (odes) in n-dimensions to a Hamiltonian form is known as the Hamiltonization problem. A solution to this problem is found in the Lie-Königs theorem which aims to solve the set of odes. In 1992, Volker Perlick (see ref. [3]), generalized the theorem using using invariant differential geometrical terminology, but his results required, as do those of Lie and Königs, the complete solution to the set of odes. It was in 1996 (see ref. [12]) when Sergio Hojman essentially changed the situation of the Hamiltonization problem by proposing a solution that does not require a complete solution to the set of odes, thereby relaxing the conditions of the Lie-K\"{o}nigs theorem. In this paper, we generalize the Hojman framework to solve the local Hamiltonization problem without a full explicit solution for the set of odes, and we construct a new Poisson tensor for the harmonic oscillator which was not obtained by Hojman.