Rédei permutations with the same cycle structure

Let $\mathbb{F}_q$ be the finite field of order $q$, and $\mathbb P^1(\mathbb{F}_q) = \mathbb F_q\cup \{\infty\}$. Write $(x+\sqrt y)^m$ as $N(x,y)+D(x,y)\sqrt{y}$. For $m\in\mathbb N$ $a \in \mathbb{F}_q$, R\'edei function $R_{m,a}\colon P^1(\mathbb F_q) \to F_q)$ is defined by $N(x,a)/D(x,a)$ if $D(x,a)\neq 0$ $x\neq\infty$, $\infty$, otherwise. In this paper we give a complete characterization all pairs $(m,n)\in\mathbb N^2$ such that permutations $R_{m,a}$ $R_{n,b}$ have same cycle structure when $a$ $b$ quadratic character $q$ odd. We explore some relationships between $(m,n)$, provide explicit families with structure. When permutation has unique not shared any other permutation, call it isolated. show only isolated are involutions. Moreover, our results can transferred to bijections form $mx$ $x^m$ on certain domains.