Jacob's ladders and the asymptotically approximate solutions of a nonlinear diophantine equation

The nonlinear equation which is connected with the main term of the Hardy-Littlewood formula for ${\zeta }^2(1\mathrm{/}2+it)$ is studied. In this direction I obtain the fine results which cannot be reached by published methods of Balasubramanian, Heath-Brown and Ivic in the field of the Hardy-Littlewood integral.