$x^{3}+y^{3}+3x^{2}-3y^{2}-3xy+6x=0 \Leftrightarrow (x+1)^{3}+(y-1)^{3}-3xy-3y+3x+3=3$
$\Leftrightarrow (x+1)^{3}+(y-1)^{3}-3(x+1)(y-1)=3$
$\left ( x+1;y-1 \right )=(m;n)$
$m^{3}+n^{3}-3mn=3 \Leftrightarrow (m+n)^{3}-3mn(m+n)-3mn-3=0$
$\Leftrightarrow (m+n+1)\left [ \left ( m+n \right )^{2} -(m+n)+1\right ]-3mn(m+n+1)=4$
$\Leftrightarrow (m+n+1)\left ( m^{2}-mn+n^{2}-m-n+1 \right )=4$
$\Delta 1: m+n+1=1 \Rightarrow m+n=0 \Rightarrow m^{2}-mn+n^{2}=3 \Rightarrow mn=-1 \Rightarrow (x;y)\in \left \{ (0;0),(-2;2) \right \}$
$\Delta 2 : m+n+1=-1 \Rightarrow m+n=-2 \Rightarrow m^{2}-mn+n^{2}+7=0 (loai)$
$\Delta3 :m+n+1=2 \Rightarrow m+n=1 \Rightarrow m^{2}-mn+n^{2}=2$
$\Rightarrow mn=-\frac{1}{3}(loai)$
$\Delta 4 : m+n+1=-2 \Rightarrow m+n=-3 \Rightarrow m^{2}-mn+n^{2}+6=0 (loai)$
$\Delta 5 : m+n+1=4 \Rightarrow m+n=3 \Rightarrow m^{2}-mn+n^{2}=3 \Rightarrow mn=2 \Rightarrow (x;y)\in \left \{ (0;3),(1;2) \right \}$
$\Delta 6 : m+n+1=-4 \Rightarrow m+n=-5 \Rightarrow m^{2}-mn+n^{2}+7=0 (loai)$
$(x;y)\in \left \{ (0;0) ,(2;2), (0;3),(1;2) \right \}$