The numerical treatment of ODE initial-value problems has been intensively researched. Energy-conservative algorithms are very important to dinamic systems. For the Hamiltonian system the symplectic algorithms are very effective. Powerful computers and algebraic software enable the creation of efficient numerical algorithms for solving ODE initial-value problems. In this paper, we propose an adaptive energy-conservative numerical-analytical algorithm for Hamiltonian systems. This algorithm is adaptable to initial-value problems where some quantities are preserved. The algorithm and its efficiency are presented for solving two-body and linear oscillator problems.