 Solution

The vector field is unchanged by reflection in the $x$-axis, $(x,y)\mapsto (x,-y)$; more generally, linearity of the second equation gives the scaling symmetry $(x,y)\mapsto (x,cy)$. It is also reversible under reflection in the $y$-axis:When $k=0$, $y$ is constant, so arbitrary translations $y\mapsto y+c$ are additional symmetries.