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In paper approach of double complex SUSY-transformations with not coincidentcomplex energies of transformation is developed, allowing to deform given realpotential $V_{1}$ with obtaining exact solutions. The explicit solutions of thedeformation of shape of the potential, its wave function at arbitrary energy,not coincident with energies of transformation, wave functions at the energiesof transformation are obtained, condition of keeping of continuity of thesolutions and isospectral condition are determined. Using a rectangular well offinite width with infinitely high walls as the starting $V_{1}$ with discreteenergy spectrum, by the proposed approach new types of deformation of thispotential with deformation of the energy spectrum as a whole have beenobtained. The new potential contains the rectangular well as own partial case(with simultaneous transformation of the shape of this new potential, energyspectrum, wave functions of all bound states, wave function at arbitrary energyinto corresponding characteristics of the rectangular well at needed choice ofparameters). Using null potential as the starting $V_{1}$ with continuousenergy spectrum, new form of reflectionless real potential has beenconstructed. This potential generalizes well-known reflectionless potential ofthe type $V_{\rm ref}(x) = A^{2}(1-2 {\rm sech}^{2}{Ax})$, allowing: to pulldown tails of the potential $V_{\rm ref}$ in the asymptotic regions up to zero(with keeping of nonzero depth); to pull down continuously the depth of thehole; to displace arbitrary along axis $x$ the hole with its passing throughzero; to create and to increase the second hole, transforming $V_{\rm ref}$into double-well potential; to control continuously and simply the asymmetry ofthe shape of such reflectionless potential.

Double complex SUSY-transformations: deformations of real potentials and their spectral characteristics