We investigate vacuum generalized Brans–Dicke theory with scalar-dependent coupling $\omega (\varphi )$ in an Einstein–Rosen-type metric. Let $W$ be the area density of the two-dimensional Killing orbits, $\varphi$ the Brans–Dicke scalar, and $\Omega =W\varphi$ their composite area density. On regions where the gradient of $\Omega$ is non-null, we impose $W=W(\varphi )$ and local invertibility of $\Omega (\varphi )$. The scalar equation admits the exact first integral $(3+2\omega )[(\Omega /\varphi )(d\varphi /d\Omega ){]}^2=\kappa$, where $\kappa$ is a signed integration constant, giving direct and inverse reconstruction of the coupling. For $\varphi =b(\Omega /{\Omega }_{*}{)}^a{e}^{q\Omega }$ one obtains $3+2\omega =\kappa /(a+q\Omega {)}^2$. The choices $\Omega =\rho$ and $\Omega =t$ define the spacelike- and timelike-gradient branches, each yielding a closed exact solution family. Their mode functions obey the cylindrical Einstein–Rosen and Euler–Poisson–Darboux equations, respectively. In the Einstein frame the normalized scalar is $\sigma ={\sigma }_0+\sqrt{|\kappa |}ln(\Omega /{\Omega }_0)$, while $q$ cancels from the metric. In the spacelike branch the scalar contributes to the radial C-energy profile separately from the tensor flux, and the reconstructed smooth-mode class has no curvature-regular axis. In the static subclass, $q=0$ reproduces the circular-speed law of a singular isothermal dark-matter halo, while $q<0$ gives a local correspondence with the rising inner NFW rotation curve. The homogeneous timelike subclass is locally Bianchi I, reduces in the Einstein frame to an exact scalar–Kasner geometry, and has a Jordan-frame effective Planck mass that evolves monotonically with Jordan-frame comoving proper time on each connected time interval on which the reconstructed scalar profile is locally invertible.