Abstract <p>A model of the supersymmetric oscillator is considered which is based on a nonlinear differential-delay equation proposed by the author in 1992. This equation provides a realization of the Heisenberg–Weyl algebra by the mixed differential-difference operators and leads to a deformation of the Chebyshev–Hermite orthogonal polynomials. We discuss general solution of this equation in a series form and describe its particular explicit solution associated with the one-gap Lamé equation.</p>