IsomorphismProblem

(1) Let $(N,\Phi)$ be a Ferrero pair with finite $N$. Let $f:N\to N$ be a group automorphism. Then $f$ is an automorphism of $(N,\mathcal B_\Phi)$ if and only if $f$ normalizes $\Phi$ in $\mathrm{Aut}(N,+)$. (Some condition on order of $N$ is required.)

(2) Let $(N,\Phi)$ be a Ferrero pair with finite $N$. Let $S_1$ and $S_2$ be two complete sets of orbit representatives of $\Phi$ in $N^*=N\setminus\{0\}$. Let $*_1$ and $*_2$ be the multiplications on $N$ induced by $S_1$ and $S_2$ respectively. Then an automorphism of $(N,+)$ is an isomorphism of $(N,+,*_1)$ and $(N,+,*_2)$ if and only if $f$ normalizes $\Phi$.

Questions: Is there any connection between these two theorems?

Question: Classify the planar nearrings defined on the Euclidean space $\mathbb R^2$.