  We investigate a Jordan-Brans-Dicke (JBD) scalar field, $\Phi$, with
power-law potential in the presence of a second scalar field, $\phi$, with an
exponential potential, in both the Jordan and the Einstein frames. We present
the relation of our model with the induced gravity model with power-law
potential and the integrability of this kind of models is discussed when the
quintessence field $\phi$ is massless, and has a small velocity. We prove that
in JBD theory, the de Sitter solution is not a natural attractor but an
intermediate accelerated solution of the form $a(t)\simeq e^{\alpha_1
t^{p_1}}$, as $t\rightarrow \infty$ where $\alpha_1>0$ and $0<p_1<1$, for a
wide range of parameters. Furthermore, in the Einstein frame we get that the
attractor is also an intermediate accelerated solution of the form
$\mathfrak{a}(\mathfrak{t})\simeq e^{\alpha_2 \mathfrak{t}^{p_2}}$ as
$\mathfrak{t}\rightarrow \infty$ where $\alpha_2>0$ and $0<p_2<1$, for the same
conditions on the parameters as in the Jordan frame. In the special case of a
quadratic potential in the Jordan frame, or for a constant potential in the
Einstein's frame, these solutions are of saddle type. Finally, we present a
specific elaboration of our extension of the induced gravity model in the
Jordan frame, which corresponds to a linear potential of $\Phi$. The dynamical
system is then reduced to a two dimensional one, and the late-time attractor is
linked with the exact solution found for the induced gravity model. In this
example the intermediate accelerated solution does not exist, and the attractor
solution has an asymptotic de Sitter-like evolution law for the scale factor.
Apart from some fine-tuned examples such as the linear, and quadratic potential
${U}(\Phi)$ in the Jordan frame, it is true that intermediate accelerated
solutions are generic late-time attractors in a modified JBD theory.
