The paper introduces a new completely integrable dispersive shallow water equation that is biHamiltonian, possessing an infinite number of conservation laws. This equation is derived using an asymptotic expansion in the Hamiltonian for Euler's equations in the shallow water regime. The soliton solutions of this equation have a limiting form with discontinuities in the first derivative at their peaks, known as "peakons." The peakons dominate the solution of the initial value problem when the constant \(\kappa\) approaches zero. The equation retains higher-order terms in the small amplitude expansion of incompressible Euler's equations, leading to an extension of the Benjamin-Bona-Mahoney (BBM) equation or the Korteweg-de Vries (KdV) equation. The biHamiltonian property allows the equation to be expressed as a compatibility condition for a linear isospectral problem, enabling the solution of the initial value problem by the inverse scattering transform (IST) method. The paper also discusses the steepening of solutions at inflection points and the dynamics of N-soliton solutions, including phase shifts and the behavior of solitons during collisions. Numerical simulations confirm the robustness of the peaked soliton solutions and the inflection point mechanism for the breakup of localized initial conditions.The paper introduces a new completely integrable dispersive shallow water equation that is biHamiltonian, possessing an infinite number of conservation laws. This equation is derived using an asymptotic expansion in the Hamiltonian for Euler's equations in the shallow water regime. The soliton solutions of this equation have a limiting form with discontinuities in the first derivative at their peaks, known as "peakons." The peakons dominate the solution of the initial value problem when the constant \(\kappa\) approaches zero. The equation retains higher-order terms in the small amplitude expansion of incompressible Euler's equations, leading to an extension of the Benjamin-Bona-Mahoney (BBM) equation or the Korteweg-de Vries (KdV) equation. The biHamiltonian property allows the equation to be expressed as a compatibility condition for a linear isospectral problem, enabling the solution of the initial value problem by the inverse scattering transform (IST) method. The paper also discusses the steepening of solutions at inflection points and the dynamics of N-soliton solutions, including phase shifts and the behavior of solitons during collisions. Numerical simulations confirm the robustness of the peaked soliton solutions and the inflection point mechanism for the breakup of localized initial conditions.