Arithmetic of pall orders and representation of integers by some ternary quadratic forms

The discrete ergodic method (Yu. V. Linnik, Izv. Akad. Nauk SSSR, Ser. Mat.,4, 363–402 (1940); A. V. Malyshev, Tr. Mat. Inst. Akad. Nauk SSSR,65) is applied to the study of properties of integral points on the ellipsoids $$\sum\nolimits_{g,m} : g(x) = m,x = (x_1 ,x_2 ,x_3 ), g(x) = \bar f(Cx),$$ where\(\bar f\) is the adjoint of one of the 39 quadratic forms of Pall (Trans. Am. Math. Soc,59, 280–332 (1946);C is an integral matrix,¦detC¦⩾1. We construct a flow of integral points on the genus surface of the ellipsoidsg(x)=m. The ergodicity of this flow and a mixing theorem are proved. We obtain an asymptotic formula for the number of representations ofm belonging to a given domain on the ellipsoid and lying in a given residue class.