  A few years ago Selivanova gave an existence proof for some integrable
models, in fact geodesic flows on two dimensional manifolds, with a cubic first
integral. However the explicit form of these models hinged on the solution of a
nonlinear third order ordinary differential equation which could not be
obtained. We show that an appropriate choice of coordinates allows for
integration and gives the explicit local form for the full family of integrable
systems. The relevant metrics are described by a finite number of parameters
and lead to a large class of models on the manifolds ${\mb S}^2, {\mb H}^2$ and
$P^2({\mb R})$ containing as special cases examples due to Goryachev,
Chaplygin, Dullin, Matveev and Tsiganov.
