Sets of lattice cubature rules that are optimal in terms of the number of nodes and exact on trigonometric polynomials in three variables

Sets of lattice cubature rules with the lattice of nodes $\lambda_k=M_k^\perp$, where the lattice $M_k$ is generated by the matrix $kB+C$ ($B$ and $C$ are integer square matrices of order $n$ independent of $k$ and $\det(B)\ne 0$) are considered. At $n=3$, for each integer $r$ ($-4\le r\le 1$), the set $S^{(\min)}$ with the trigonometric $(6k+r)$ property and the asymptotically minimal number of nodes $N^{(\min)}(k)$ is found. This means that, for any set $S^{(\min)}$ with the trigonometric $(6k+r)$ property and the number of nodes $N(k)$, the inequality $N(k)\ge N^{(min)}(k)$ holds true if $k$ is sufficiently large. Certain properties of the optimal sets $S^{(min)}$ and the nearest (in terms of the number of nodes) sets $S^{(\min+)}$ are investigated.