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In this work, we introduce a new realization of exactly-solvabletime-dependent Hamiltonians based on the solutions of the fourth Painlev\'e andthe Ermakov equations. The latter is achieved by introducing a shape-invariantcondition between an unknown quantum invariant and a set of third-orderintertwining operators with time-dependent coefficients. The new quantuminvariant is constructed by adding a deformation term to the well-knownparametric oscillator invariant. Such a deformation depends explicitly on timethrough the solutions of the Ermakov equation, which ensures the regularity ofthe new time-dependent potential of the Hamiltonian at each time. On the otherhand, with the aid of the proper reparametrization, the fourth Painlev\'eequation appears, the parameters of which dictate the spectral behavior of thequantum invariant. In particular, the eigenfunctions of the third-order ladderoperators lead to several sequences of solutions to the Schr\"odinger equation,determined in terms of the solutions of a Riccati equation, Okamotopolynomials, or nonlinear bound states of the derivative nonlinearSchr\"odinger equation. Remarkably, it is noticed that the solutions in termsof the nonlinear bound states lead to a quantum invariant with equidistanteigenvalues, which contains both an (N+1)-dimensional and an infinite sequenceof eigenfunctions. The resulting family of time-dependent Hamiltonians is suchthat, to the authors' knowledge, have been unnoticed in the literature ofstationary and nonstationary systems.

Fourth Painlev\'e and Ermakov equations: quantum invariants and new exactly-solvable time-dependent Hamiltonians