Integrable Cosmological Potentials
  The problem of classification of the Einstein--Friedman cosmological
Hamiltonians $H$ with a single scalar inflaton field $\varphi$ that possess an
additional integral of motion polynomial in momenta on the shell of the
Friedman constraint $H=0$ is considered. Necessary and sufficient conditions
for the existence of first, second, and third degree integrals are derived.
These conditions have the form of ODEs for the cosmological potential
$V(\varphi)$. In the case of linear and quadratic integrals we find general
solutions of the ODEs and construct the corresponding integrals explicitly. A
new wide class of Hamiltonians that possess a cubic integral is derived. The
corresponding potentials are represented in a parametric form in terms of the
associated Legendre functions. Six families of special elementary solutions are
described and sporadic superintegrable cases are discussed.
