  We consider magnetic billiards under a strong constant magnetic field. The
purpose of this paper is two-folded. We examine the question of existence of
polynomial integral of billiard magnetic flow. We succeed to reduce this
question to algebraic geometry test on existence of polynomial integral, which
shows polynomial non-integrability for all but finitely many values of the
magnitude. In the second part of the paper we construct examples of magnetic
billiards which have the so called $\delta$-Gutkin property, meaning that any
Larmor circle entering the domain with angle $\delta$ exits the domain with the
same angle $\delta$. For ordinary Birkhoff billiard in the plane such examples
were introduced by E. Gutkin and are very explicit. Our construction of Gutkin
magnetic billiards relies on beautiful examples by F.Wegner of the so called
Zindler curves, which are related to the problem of floating bodies in
equilibrium, which goes back to S.Ulam. We prove that Gutkin magnetic billiard
can be obtained as a parallel curve to a Wegner curve. Wegner curves can be
written by elliptic functions in polar coordinates so the construction of
magnetic Gutkin billiard is rather explicit but much more complicated.
