There is the system:
$$ \left\{ \begin{array}{c} \dot{x} = -xy^2+x+y \\ \dot{y} = -x-y+x^2y \end{array} \right. $$
the way it should be solved is to find an integrable combination there is the description (page 349 of pdf document).
I've already tried multiplying the first equation by $x$, the second one by $y$ then adding first to second so I got: $xdx+ydy = x^2 - y^2$, which I have no idea how to integrate.
Also I tried multiplying the first by $y$, the second by $x$, also adding first to second so I got: $ydx+xdy=(xy - 1)(x^2 - y^2)$, which I also don't know how to integrate.
Could you plese provide any integrating combinations or ideas how to deal with equations I got.
That's a good start, except that you should write $\dot x$ and $\dot y$ instead of $dx$ and $dy$.
Anyway, you have $$ (\tfrac12 (x^2+y^2))\dot{} = x \dot x + y \dot y = x^2-y^2 $$ and $$ (xy)\dot{} = \dot x y + x \dot y = (xy-1)(x^2-y^2) . $$ These can be combined to give $$ (xy)\dot{} = (xy-1) (\tfrac12 (x^2+y^2))\dot{} $$ so that either $xy-1=0$ identically or $$ \frac{(xy)\dot{}}{xy-1} = (\tfrac12 (x^2+y^2))\dot{} $$ where both sides can be integrated to find a constant of motion.
Can you take it from there, or do you need further hints?