  We consider a spatially flat Friedmann--Lema\^{\i}tre--Robertson--Walker
background space with an ideal gas and a multifield Lagrangian consisting of
two minimally coupled scalar fields which evolve in a field space of constant
curvature. For this cosmological model we classify the potential function for
the scalar fields such that variational point symmetries exist. The
corresponding conservation laws are calculated. Finally, analytic solutions are
presented for specific functional forms of the scalar field potential in which
the cosmological field equations are characterized as a Liouville integrable
system by point symmetries. The free parameters of the cosmological model are
constrained in order to describe analytic solutions for an inflationary epoch.
Finally, stability properties of exact closed-form solutions are investigated.
These solutions are scaling solutions with important physical properties for
the cosmological model.
