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In this paper, without any assumption on $v$ and under extremely mildassumption $u(x)=O(|x|^{K})$ at $\infty$ for some $K\gg1$ arbitrarily large, weprove classification of solutions to the following conformally invariant systemwith mixed order and exponentially increasing nonlinearity in $\mathbb{R}^{2}$:\begin{equation*}\\\begin{cases} (-\Delta)^{\frac{1}{2}}u(x)=e^{pv(x)}, \qquadx\in\mathbb{R}^{2}, \\ -\Delta v(x)=u^{4}(x), \qquad x\in\mathbb{R}^{2},\end{cases}\end{equation*} where $p\in(0,+\infty)$, $u\geq 0$ and satisfies thefinite total curvature condition$\int_{\mathbb{R}^{2}}u^{4}(x)\mathrm{d}x<+\infty$. In order to show integralrepresentation formula and crucial asymptotic property for $v$, we derive anduse an $\exp^{L}+L\ln L$ inequality, which is itself of independent interest.When $p=\frac{3}{2}$, the system is closely related to single conformallyinvariant equations $(-\Delta)^{\frac{1}{2}}u=u^{3}$ and $-\Delta v=e^{2v}$ on$\mathbb{R}^{2}$, which have been quite extensively studied (cf.\cite{BF,C,CY,CL,CLL,CLZ} etc). We also derive classification results fornonnegative solutions to conformally invariant system with mixed order andHartree type nonlocal nonlinearity in $\mathbb{R}^{3}$. Extensions to mixedorder conformally invariant systems in $\mathbb{R}^{n}$ with general dimensions$n\geq3$ are also included.

Classification of solutions to conformally invariant systems with mixed order and exponentially increasing or nonlocal nonlinearity