| Summary: | The <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>4</mn><mo>×</mo><mn>4</mn></mrow></semantics></math></inline-formula> trace-free complex matrix set is introduced in this study. By using it, we are able to create a novel soliton hierarchy that is reduced to demonstrate its bi-Hamiltonian structure. Additionally, we give the Darboux matrix T, whose elements are connected to the spectral parameter in accordance with the various positions and numbers of the spectral parameter <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>λ</mi></semantics></math></inline-formula>. The Darboux transformation approach has also been successfully applicated to superintegrable systems.
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