Solution
= 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:
$$
(x(t),y(t))\longmapsto(-x(-t),y(-t)).
$$
When $k=0$, $y$ is constant, so arbitrary translations $y\mapsto y+c$ are additional symmetries.