Entire solutions of system of Fermat-type difference and partial differential-difference equations in ${\mathbb{C}}^2$

In this paper we mainly study the existence and the form of entire solutions with finite order for the following system of Fermat-type difference and partial differential-difference equations $$\{\begin{array}{l}{f}_1(z{)}^2+({\mathrm{\Delta }}_c{f}_2(z){)}^2=1\\ f_2(z{)}^2+({\mathrm{\Delta }}_c{f}_1(z){)}^2=1,\end{array}$$ $$\{\begin{array}{l}{a}_1^2{f}_1(z{)}^2+(a_2{f}_2(z+c)+a_3{f}_2(z){)}^2=1\\ a_1^2{f}_2(z{)}^2+(a_2{f}_1(z+c)+a_3{f}_1(z){)}^2=1,\end{array}$$ $$\{\begin{array}{l}(a_1{f}_1(z+c)+a_2{f}_1(z){)}^2+(a_3{f}_2(z+c)+a_4{f}_2(z){)}^2=1\\ (a_1{f}_2(z+c)+a_2{f}_2(z){)}^2+(a_3{f}_1(z+c)+a_4{f}_1(z){)}^2=1,\end{array}$$ and $$\{\begin{array}{l}({\mathrm{\partial }}^I{f}_1(z)+{\mathrm{\partial }}^J{f}_1(z){)}^{n_1}+f_2(z+c{)}^{m_1}=1\\ ({\mathrm{\partial }}^I{f}_2(z)+{\mathrm{\partial }}^J{f}_2(z){)}^{n_2}+f_1(z+c{)}^{m_2}=1\end{array}$$ in several complex variables. Some of our results are improvements and extensions of the previous theorems given by Zheng-Xu \cite{Zheng-Xu & Analysis math & 2021}, Xu-Cao \cite{Xu & Cao & 2018}, Xu $\mathit{\text{et.}}\mathit{\text{al.}}$ \cite{Xu-Liu-Li-JMAA-2020} and Li $\mathit{\text{et.}}\mathit{\text{al.}}$ \cite{Li-Zhang-Xu & 2021 & AIMS}. Moreover, we give some examples which are relevant to the content of the paper.