 Solution

Write $r_a={\mu }_{0,a}^2$. The stationary equations forareFor $r_1,r_2>0$, the disordered minimum is $(0,0)$. If $r_1<0$ and $r_1<r_2$, the first component orders with ${\varphi }_1^2=-r_1/(4g_0)$ and ${\varphi }_2=0$. If $r_2<0$ and $r_2<r_1$, the second component orders analogously.

The positive $r_2$ half of $r_1=0$ and the positive $r_1$ half of $r_2=0$ are continuous-transition lines. Along the negative diagonal $r_1=r_2<0$, the potential has an enhanced $O(2)$ symmetry and a circle of minima. Crossing that diagonal exchanges the two ordered axes and makes derivatives of the minimum free energy jump, so it is a first-order phase transition line. The origin, where the two continuous lines meet the first-order line, is a bicritical point.