The Algebraic Cluster Model(ACM) is an interacting boson model that gives the relative motion of the cluster configurations in which all vibrational and rotational degrees of freedom are present from the outset. We schemed a solvable extended transitional Hamiltonian based on the SU(1;1) Lie algebra within the framework for two-, three- and four-body algebraic cluster models that explains both regions O(4) U(3), O(7) U(6) and O(10)-U(9), respectively. We suggest that this method can be used to study of k + x nucleon structures with k = 2, 3, 4 and x = 1, 2 and so on. The obtained results in this study confirm that this ACM technique is worth extending for investigating odd-A and odd-odd nuclei. So, the clustering survives the addition of one and two particles. Our studies confirm the importance of the odd nuclei as necessary signatures to characterize the occurrence of the phase transition and to determine the precise position of the critical point.