The authors present a complete study of quadratic 3-parameter unfoldings of the integrable quadratic system \[ \dot x=- y-3x^2+ y^2,\;\dot y= x(1-2y) \leqno (1)_0 \] belonging to the class \(Q^{\mathbb{R}_3}\) (according to H. Zoladek), or the class \(m=a=0\) (according to Chinese classification). System \((1)_0\) has two centers \((0,0)\) and \((0,1)\), one invariant line: \(L_1=1- 2y=0\), one invariant hyperbola: \(L_2= 6x^2- 6y^2+ 6y-1=0\) and the general integral: \(L_1^{-3} L_2=C\). The following theorem is proved: Theorem. i) The 3-parameter family: \[ \dot x=-y- 3x^2+y^2+ \mu_1x+ \mu_2xy, \quad \dot y=x(1-2y)+ \mu_3x^2 \leqno (1)_\mu \] (where \(\mu= (\mu_1, \mu_2, \mu_3)= (\delta\nu_1, \delta \nu_2, \delta \nu_3)\) with \(0<\delta\ll 1)\) is a versal unfolding of \((1)_0\), among all 3-parameter unfoldings of \((1)_0\), transverse to the stratum \(Q^{\mathbb{R}}_3\). ii) The bifurcation diagram of \((1)_\mu\) has a conic structure in \(\mathbb{R}^3\) for \(0< \delta\ll 1\), it is point-symmetric w.r.t. \(\mu=0\). Hence it can be expressed (for \(\mu\neq 0)\) by drawing its intersection with the half sphere \(S^+_\gamma =\{ \sum^3_{i=1} \mu^2_i= \gamma^2\), \(\mu_3\geq 0: 0<\gamma\ll 1\}\). iii) The intersection of the bifurcation diagram with \(S^+_\gamma\) and the related 17 structurally stable phase-portraits are shown clearly. In which the total number of limit cycles is at most 3; but if they are distributed around two foci, then the distribution must be (1,1).