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The construction of any finite translation plane depends on the appropriate determination of a partition $P\left(E\right)$ of a Galois field $E$, together with a set $A$ of automorphisms of $E$ as a vector space. In this paper we obtain sufficient conditions on $P\left(E\right)$ and $A$, so that a translation plane is produced. They are also necessary conditions when $\left|A\right|=1$. Particularly, we examine the case where $E$ is a two-dimensional vector space. We prove that no translation planes are constructible by a single automorphism, other than...