$y = x-1-\dfrac {2010} {x+3}$, so we need $x+3$ to be a divisor of $2010 = 2\cdot 3\cdot 5\cdot 67$. In the same time we need $(x-2)(x-3)\geq 2010$, hence $x>45$. All these leave very few cases.

$x^2-xy+2x-3y=2013\Leftrightarrow (x+3)(x-y-1)=2\cdot 3\cdot 5\cdot 67$, $\Rightarrow (x,y)=(64,33),(131,115),(298,287),(332,325),(399,393),(667,663),(1002,999),(2007,2005)$.