The higher-order Henneberg-type minimal surfaces family in $\mathbb{R}^4$

We consider a higher-order Henneberg-type minimal surfaces family using the generalized Weierstrass--Enneper representation in four-dimensional space $\mathbb{R}^4$. We derive explicit parametric equations for the surface and determine its differential geometric characteristics, including the normal vector fields $\mathbf{n}_1$ and $\mathbf{n}_2$, as well as the Gauss curvature. Furthermore, by projecting these parametric forms from four to three dimensions, we generate visualizations that reveal the geometric structure of the Henneberg-type minimal surface. In addition, we examine the integral-free form and derive the corresponding algebraic function for this family of surfaces.