  We generalize, to some extent, the results on integrable geodesic flows on
two dimensional manifolds with a quartic first integral in the framework laid
down by Selivanova and Hadeler. The local structure is first determined by a
direct integration of the differential system which expresses the conservation
of the quartic observable and is seen to involve a finite number of parameters.
The global structure is studied in some details and leads to a class of models
living on the manifolds S^2, H^2 or R^2. As special cases we recover
Kovalevskaya's integrable system and a generalization of it due to Goryachev.
