There are solutions, namely $(0,1),(0,-1)$. To find all solutions we may restrict ourselves to the first quadrant, since the equations only involve $|x|$ and $|y|$. Then $2x+y=1 \implies x\in [0,1/2], y\in [0,1] \implies [ x]=0 \implies [ 2y ]=2 \implies y\ge 1 \implies y=1\implies (x,y)=(0,1)$.