= Solution
Let $x^*$ maximize SYSTEM. Under the usual positive-capacity, nonempty-route assumptions, the feasible set is compact, contains a positive point, and has a <Slater condition> point. Strict <concavity> makes the optimum unique. The condition $U_r'(0)=\infty$ forces $x_r^*>0$: moving from any optimum with a zero component slightly toward a positive feasible point gives an infinite positive directional derivative in that component and only finite losses in the positive components.
By the <Karush-Kuhn-Tucker conditions>, there are link <Lagrange multipliers> $p_j^*\geq0$ with
$$
U_r'(x_r^*)=(A^Tp^*)_r,\qquad p_j^*\bigl(C_j-(Ax^*)_j\bigr)=0.
$$
Set the <route congestion price> $y_r^*=(A^Tp^*)_r=U_r'(x_r^*)$ and choose
$$
\boxed{\nu_r^*=x_r^*e^{y_r^*}.}
$$
Then $x_r^*=\nu_r^*e^{-y_r^*}$, as required. For USER with fixed $y_r^*$, the substitution $x=\nu_r e^{-y_r^*}$ is a bijection of the nonnegative half-line, and its objective becomes $U_r(x)-y_r^*x$. This is a <strictly concave function> whose derivative vanishes at $x_r^*$, so its unique maximizer is $x_r^*$ and the corresponding USER optimizer is $\nu_r^*$.
For NETWORK at $\nu^*$, its objective derivative at $x^*$ is
$$
\log\frac{\nu_r^*}{x_r^*}=y_r^*=(A^Tp^*)_r.
$$
Thus $x^*,p^*$ obey its <Karush-Kuhn-Tucker conditions>, with the same feasibility and <complementary slackness>. The NETWORK objective is a <strictly concave function>, so $x^*$ is also its unique maximizer. \b[The triple $(x^*,y^*,\nu^*)$ solves all three problems simultaneously.] This is a <user-network decomposition of concave utility maximization>.
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