  An endo-commutative algebra is a nonassociative algebra in which the square
mapping preserves multiplication. In this paper, we give a complete
classification of 2-dimensional endo-commutative straight algebras of rank one
over an arbitrary non-trivial field, where a straight algebra of dimension 2
satisfies the condition that there exists an element $x$ such that $x$ and
$x^2$ are linearly independent. We list all multiplication tables of the
algebras up to isomorphism.
