Construct a sequence of lattice rules for the upper estimation of the value \[ M( P(S_\gamma (n)))= \sup\{|t(\cdot) |_\infty/ |t(\cdot) |_1: t\in P(S_\gamma (n)),\;t\not\equiv 0\} \] for \(n=2\). \(P(S_\gamma (n))\) denotes the class of all nonnegative trigonometric polynomials \(t(x)\) with the spherical spectrum \(S_\gamma(n)\). The main result of this paper is the estimate \[ {\textstyle {\pi \over 4}}+ o(1)\leq M(P (S_\gamma (2)))/ \gamma^2\leq {\textstyle {15 \over 17}}+ o(1) \qquad (\gamma\to \infty). \] The estimate for \(n=3\) has been published in 1993.