  In this paper we prove the following result: if two 2-dimensional
2-homogeneous rational vector fields commute, then either both vector fields
can be explicitly integrated to produce rational flows with orbits being lines
through the origin, or both flows can be explicitly integrated in terms of
algebraic functions. In the latter case, orbits of each flow are given in terms
of $1$-homogeneous rational functions $W$ as curves $W(x,y)=\textrm{const}$. An
exhaustive method to construct such commuting algebraic flows is presented. The
degree of the so-obtained algebraic functions in two variables can be
arbitrarily high.
