  The derivation of conservation laws and invariant functions is an essential
procedure for the investigation of nonlinear dynamical systems. In this study
we consider a two-field cosmological model with scalar fields defined in the
Jordan frame. In particular we consider a Brans-Dicke scalar field theory and
for the second scalar field we consider a quintessence scalar field minimally
coupled to gravity. For this cosmological model we apply for the first time a
new technique for the derivation of conservation laws without the application
of variational symmetries. The results are applied for the derivation of new
exact solutions. The stability properties of the scaling solutions are
investigated and criteria for the nature of the second field according to the
stability of these solutions are determined.
