Jacob's ladders, our old formula (1985) and new $\zeta$-equivalent of the Fermat-Wiles theorem on two-parametric set of lemniscates of Bernoulli

In our paper from 1985 we have constructed two integrals of the Riemann’s function $Z^2(t)$ over two disconnected sets with asymptotically equal measures such that these two integrals differ by considerably big excess. In the present paper we use the formula for that excess to construct a new $\zeta$-equivalent of the Fermat-Wiles theorem on a two-parametric set of lemniscates of Bernoulli.