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The building blocks of the interacting boson model (IBM) are associated with both s and d bosons for positive parity states. An extension of sd−IBM along these models to spdf−IBM can provide the appropriate framework to describe negative parity states. In this paper, a solvable extended transitional Hamiltonian based on the affine ( )  SU 1, 1 Lie algebra is proposed to describe low lying positive and negative parity states between the spherical and deformed gamma-unstable shape. Quantum phase transitions (QPTs) are investigated based on dual algebraic structures for the four-level pairing model. Numerical extraction to low-lying energy levels and transition rates within the control parameters of this evaluated Hamiltonian are presented for various N values. By reproducing the experimental results, the method based on the signatures of the phase transition, such as the expectation value of the boson number operators in the lowest excited states, are used to provide a better description of Ru isotopes in this transitional region.

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