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The procedure proposed recently by J.Bougie, A.Gangopadhyaya and J.V.Mallowto study the general form of shape invariant potentials in one-dimensionalSupersymmetric Quantum Mechanics (SUSY QM) is generalized to the case of HigherOrder SUSY QM with supercharges of second order in momentum. A new shapeinvariant potential is constructed by this method. It is singular at theorigin, it grows at infinity, and its spectrum depends on the choice ofconnection conditions in the singular point. The corresponding Schr\"odingerequation is solved explicitly: the wave functions are constructed analytically,and the energy spectrum is defined implicitly via the transcendental equationwhich involves Confluent Hypergeometric functions.

New Implicitly Solvable Potential Produced by Second Order Shape Invariance