Implicit equations of the henneberg-type minimal surface in the four dimensional euclidean space

Considering the Weierstrass data as $(\psi ,f,g)=( 2,1-z^{-m},z^{n})$, we introduce a two parameter family of Henneberg type minimal surface that we call $\mathfrak{H}_{m,n}$ for positive integers $(m,n)$ by using the Weierstrass representation in the four-dimensional Euclidean space $\mathbb{E}^{4}$. We define $\mathfrak{H}_{m,n}$ in $(r,\theta)$ coordinates for positive integers $(m,n)$ with $m\neq 1,n\neq -1, -m+n\neq -1$, and also in $(u,v)$ coordinates, and then we obtain implicit algebraic equations of the Henneberg type minimal surface of values $(4,2)$.