I'm having trouble with solving problems conserning the symmetries of mathematical objects. The task is to find what the symmetry transformations are and write
down the symmetry group they form (the allowed transformations are rotations, refections and translations, of both space and time:


The vector field: (x,y)\mapsto \vec{e{y}} where \vec{e{y}} is a vector e with a subscript y
The vector field: (x,y)\mapsto x\vec{e{x} } + y*\vec{e{y}}
The vector fi eld: (x,y)\mapsto y*\vec{e{x}} - x*\vec{e{y}}

The one-dimensional equation of motion = u + a for some fixed real number a, where =x" w.r.t. time
The one-dimensional equation of motion = a*u for some fixed real number a where =x' w.r.t. time
The one-dimensional equation of motion = 3 where =x' w.r.t. time