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 $\mathrm{\Omega }=W\varphi$ their composite area density. On regions where the gradient of $\mathrm{\Omega }$ is non-null, we impose $W=W(\varphi )$ and local invertibility of $\mathrm{\Omega }(\varphi )$. The scalar equation admits the exact first integral $(3+2\omega )[(\mathrm{\Omega }/\varphi )(d\varphi /d\mathrm{\Omega }){]}^2=\kappa$, where $\kappa$ is a signed integration constant, giving direct and inverse reconstruction of the coupling. For $\varphi =b(\mathrm{\Omega }/{\mathrm{\Omega }}_{*}{)}^a{e}^{q\mathrm{\Omega }}$ one obtains $3+2\omega =\kappa /(a+q\mathrm{\Omega }{)}^2$. The choices $\mathrm{\Omega }=\rho$ and $\mathrm{\Omega }=t$ 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 $\sigma ={\sigma }_0+\sqrt{|\kappa |}\ln (\mathrm{\Omega }/{\mathrm{\Omega }}_0)$, 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.

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• Alternative gravity theories
• General relativity
• Gravitational waves