Geodesic Flows on the Klein Bottle, Integrable by Polynomials in Momenta of Degree Four

In the present paper we construct and topologically describe a series of examples of metrics on the Klein bottle such that for each metric
$ \bullet $ the corresponding geodesic flow has an integral, which is a polynom of degree four in momenta
$ \bullet $ the corresponding geodesic flow has no integral, which is a polynom of degree less than four in momenta.