Since $x^{2} + 3xy + y^{2} = 909$ and $3x^{2} + xy + 3y^{2} = 1287$, then adding these gives $4x^{2} + 4xy + 4y^{2} = 2196$. Dividing by 4 gives $x^{2} + xy + y^{2} = 549$. Subtracting this from the first equation gives $2xy = 360$, so $xy = 180$. Substituting $xy = 180$ into the first equation gives $x^{2} + 2xy + y^{2} = 729$, which is $(x+y)^{2} = 27^{2}$. Therefore, $x+y = 27$ or $x+y = -27$. This also shows that $x+y$ cannot equal any of 39, 29, 92, and 41.