= Solution
The fixed-point equation for $u$ factors as
$$
u\bigl[(1+u^2)^2-4u\bigr]=u(u-1)(u^3+u^2+3u-1)=0.
$$
The cubic derivative is $3u^2+2u+3>0$ for every real $u$; its values at zero and one have opposite signs. Thus the possible first coordinates in the stated domain are exactly $u=0,u_*,1$.
For finite $v$, the second fixed-point equation is $v[v^3-(1+u^2)^2]=0$. The <renormalization-group fixed points> are consequently
$$
\boxed{(a,0),\quad \bigl(a,(1+a^2)^{2/3}\bigr),\quad (a,\infty),\qquad a\in\{0,u_*,1\}.}
$$
There are six finite points and three boundary points at infinity. Infinity is interpreted in a compactified domain, not as an ordinary real number. The <reciprocal coordinate at infinite coupling> $w=1/v$ transforms as $w'=(1+u^2)^2w^4$, making $w=0$ a well-defined boundary fixed point.
At any finite fixed point $(\tilde u,\tilde v)$, let $\delta u=u-\tilde u$, $\delta v=v-\tilde v$. The <Jacobian matrix> linearization is
$$
\begin{pmatrix}\delta u'\\\delta v'\end{pmatrix}=\begin{pmatrix}\dfrac{8\tilde u(1-\tilde u^2)}{(1+\tilde u^2)^3}&0\\[5pt]-\dfrac{4\tilde u\tilde v^4}{(1+\tilde u^2)^3}&\dfrac{4\tilde v^3}{(1+\tilde u^2)^2}\end{pmatrix}\begin{pmatrix}\delta u\\\delta v\end{pmatrix}+O(\|(\delta u,\delta v)\|^2).
$$
The only point with both coordinates strictly interior is $\tilde u=u_*$, $\tilde v=(1+u_*^2)^{2/3}$. Using its fixed-point relations, the <matrix> becomes
$$
M_* =\begin{pmatrix}\rho&0\\c&4\end{pmatrix},\qquad \rho=\frac{2(1-u_*^2)}{1+u_*^2},\qquad c=-\frac{4u_*\tilde v}{1+u_*^2}.
$$
Its <eigenvalues> are $\rho$ and $4$. In particular, $u_*<1/\sqrt3$ because the increasing cubic is already positive there, so $\rho>1$ (numerically about $1.68$). The corresponding <eigenvectors> can be chosen as $(1,c/(\rho-4))^T$ and $(0,1)^T$. Both discrete multipliers exceed one, hence
$$
\boxed{\text{the unique interior fixed point is repulsive in both directions}.}
$$
Both perturbations are <relevant directions of a fixed point> under repeated coarse-graining, rather than one stable and one unstable direction. A length-rescaling factor was not specified, so these multipliers should not be assigned numerical critical scaling exponents without additional information. At $a=0,1$, the $u$ multiplier is zero; at $v=0$ or $w=0$ the other multiplier is also zero. Thus the four corner points attract locally within the domain, the two positive finite boundary points at $a=0,1$ are saddles, and the points $(u_*,0),(u_*,\infty)$ have one repulsive and one attractive direction.
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