We make a study on the Poincare disc.

First we study the system on the U1 chart (x=1/z2,y=z1/z2):
    z1' = -5.*z1*z2+z1^3*z2+z1^2*z2^2+2.*z1+z2^2
    z2' = 2.0000000000*z1*z2+2.*z1^2*z2-3.*z2^2+z1^2*z2^2+z1*z2^3
The point (0.,0) is a semi hyperbolic point.
The point is a saddle-node.
First we study the system on the U2 chart (x=z1/z2,y=1/z2):
    z1' = 5.*z1^2*z2-1.*z2-1.*z2^2-2.*z1^3-1.*z1^2*z2^2
    z2' = -2.*z1^2*z2+2.0000000000*z1*z2+2.*z2+2.*z1*z2^2-1.*z1*z2^3
The point (0,0) is a semi hyperbolic point.
The point is a saddle-node.