𝒫𝒯-symmetric properties of the Toomre instability. Top panel: Curve of EPs corresponding to marginal stability. A maximum is reached for Q = 1 as expected. Bottom panel: Krein collision of the stability of the Toomre problem when varying the parameters along the black line of the top panel. Depending on the value of Q, exactly one of the two Krein quantities identified is zero, depending on which symmetry is broken by the perturbation. The curve in green is X ⋅ VX, orange is $X·\stackrel{\sim }{V}X$, solid blue is Re(ω), and dotted blue is Im(ω), for a destabilising mode.