| Summary: | We present a family of hypersurfaces of revolution distinguished by four parameters in the five-dimensional pseudo-Euclidean space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi mathvariant="double-struck">E</mi><mrow><mn>2</mn></mrow><mn>5</mn></msubsup></semantics></math></inline-formula>. The matrices corresponding to the fundamental form, Gauss map, and shape operator of this family are computed. By utilizing the Cayley–Hamilton theorem, we determine the curvatures of the specific family. Furthermore, we establish the criteria for maximality within this framework. Additionally, we reveal the relationship between the Laplace–Beltrami operator of the family and a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>5</mn><mo>×</mo><mn>5</mn></mrow></semantics></math></inline-formula> matrix.
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