The unique phenomena in acoustic metamaterial at the Dirac-like cone, and at the exceptional spawning ring could transform the field of engineering with multiple new applications that was never possible before. Formation of localized conical dispersion (Dirac cone) at the Brillouin boundaries are the well-known facts, which exhibits many intriguing phenomena. However, Dirac cone like dispersion at the center of Brillouin zone (k ⃗=0) is rare and only happens due to accidental degeneracy at finite frequencies in two-dimensional periodic crystals (PCs), with or without microarchitectures. Additionally, a possible deformation of a Dirac cone instigates a degenerated state called spawning ring or exceptional ring, where two resonant modes coincides over a zonal wave numbers. The point of exceptional ring is called exceptional point, and known as parity-time symmetry breaking point. Exploiting the behaviors of Dirac cones and spawning rings at the origin and boundaries of the Brillouin zone, a directional and bifurcation lens were designed which will propagate sound wave in specific directions at multiple frequencies. In this article, PCs having a square array of cylindrical polyvinylchloride (PVC) inclusions in air media are studied numerically, that exhibits Dirac-like points and exceptional points simultaneously at k ⃗=0 by modulating the physical parameter of the cylindrical inclusions (PVC) in fixed lattice constant. Detailed numerical study of 2D PCs showed that by adjusting the system parameter, an accidental triple degeneracy of dispersion at Γ point can be achieved. The authenticity of the claim is demonstrated by simulating the phenomena in a designed zero-refractive index material.