  In the paper, we consider the class of so-called endo-commutative algebras.
From the identity imposed to specify this class, one can easily see that the
product in this class preserves the square of elements. We give a complete
classification of this class, up to isomorphism, over any basic field in
dimension two. This elementary and self-contained exposition uses the recent
result of one of the authors on description of the category of all
$2$-dimensional algebras over any basic field.
