  In this paper, without any assumption on $v$ and under the extremely mild
assumption $u(x)= O(|x|^{K})$ as $|x|\rightarrow+\infty$ for some $K\gg1$
arbitrarily large, we classify solutions of the following conformally invariant
system with mixed order and exponentially increasing nonlinearity in
$\mathbb{R}^{3}$: $$ \begin{cases} \ (-\Delta)^{\frac{1}{2}} u=v^{4} ,&x\in
\mathbb{R}^{3},\\ \ -\Delta v=e^{pw} ,&x\in \mathbb{R}^{3},\\ \
(-\Delta)^{\frac{3}{2}} w=u^{3} ,&x\in \mathbb{R}^{3}, \end{cases} $$ where
$p>0$, $w(x)=o(|x|^{2})$ at $\infty$ and $u,v\geq0$ satisfies the finite total
curvature condition $\int_{\mathbb{R}^{3}}u^{3}(x)\mathrm{d}x<+\infty$.
Moreover, under the extremely mild assumption that \emph{either} $u(x)$ or
$v(x)=O(|x|^{K})$ as $|x|\rightarrow+\infty$ for some $K\gg1$ arbitrarily large
\emph{or} $\int_{\mathbb{R}^{4}}e^{\Lambda pw(y)}\mathrm{d}y<+\infty$ for some
$\Lambda\geq1$, we also prove classification of solutions to the conformally
invariant system with mixed order and exponentially increasing nonlinearity in
$\mathbb{R}^{4}$: \begin{align*} \begin{cases} \ (-\Delta)^{\frac{1}{2}}
u=e^{pw} ,&x\in \mathbb{R}^{4},\\ \ -\Delta v=u^2 ,&x\in \mathbb{R}^{4},\\ \
(-\Delta)^{2} w=v^{4} ,&x\in \mathbb{R}^{4}, \end{cases} \end{align*} where
$p>0$, and $w(x)=o(|x|^{2})$ at $\infty$ and $u,v\geq0$ satisfies the finite
total curvature condition $\int_{\mathbb{R}^{4}}v^{4}(x)\mathrm{d}x<+\infty$.
The key ingredients are deriving the integral representation formulae and
crucial asymptotic behaviors of solutions $(u,v,w)$ and calculating the
explicit value of the total curvature.
