  We use a mathematical approach based on the constraints systems in order to
reconstruct the equation of state and the inflationary potential for the
inflaton field from the observed spectral indices for the density perturbations
$n_{s}$ and the tensor to scalar ratio $r$. From the astronomical data, we can
observe that the measured values of these two indices lie on a two-dimensional
surface. We express these indices in terms of the Hubble slow-roll parameters
and we assume that $n_{s}-1=h\left( r\right) $. For the function $h\left(
r\right) $, we consider three cases, where $h\left( r\right) $ is constant,
linear and quadratic, respectively. From this, we derive second-order equations
whose solutions provide us with the explicit forms for the expansion
scale-factor, the scalar-field potential, and the effective equation of state
for the scalar field. Finally, we show that for there exist mappings which
transform one cosmological solution to another and allow new solutions to be
generated from existing ones.
