A new approach toward understanding the heavyfermion systems (HFS) within a framework of the almost degenerate lattice Anderson Hamiltonian in the Kondo regime is proposed. In the coherent low temperature regime operators in the effective Hamiltonian are found to belong to an SU(2J 4-3) dynamical algebra. A canonical transformation is employed to decouple the quasiparticle branches, thereby setting up the decoupling equation. It is found that this decoupling equation has a solution of the symmetry-altering type 1. The thermodynamic response functions and other quantities are calculated for this new state. This solution is a consequence of the degeneracy of the uncoupled f-orbitals. It is characterized by the interatomic hopping off-electrons, which produces the spin delocalization regime and with the renormalized f-level pinned close to the Fermi level. This is also found to be the source of the apparent spin compensation regime, which is accompanied by large enhancement of the thermodynamic response functions. In addition, the caiculated phase coherence length is found to be much greater than a lattice constant, thereby showing a many-body character of this new state. It is believed that this new state provides an accurate description of the heavy Fermion state at low temperatures. The stability conditions for the new regime are also discussed.