Two-field Cosmological $\alpha$-attractors with Noether Symmetry

We study Noether symmetries in two-field cosmological $\alpha$-attractors, investigating the case when the scalar manifold is an elementary hyperbolic surface. This encompasses and generalizes the case of the Poincare disk. We solve the conditions for the existence of a `separated' Noether symmetry and find the form of the scalar potential compatible with such, for any elementary hyperbolic surface. For this class of symmetries, we find that the $\alpha$-parameter must have a fixed value. Using those Noether symmetries, we also obtain many exact solutions of the equations of motion of these models, which were studied previously with numerical methods.