at the tunnel barrier I (*mode(1a)*) or at the ferromagnetic F-layer (*mode(1b)*). The separating black solid vertical lines in the upper part in Fig.\[Phase\] represent the locus of points where the critical currents of SIs and sFS parts of SIsFS junction are equal. The dashed lines give the locations of the points of $0$ to $\pi $ transitions, $d_{Fn}=\pi (n-3/4)\xi _{F}\sqrt{2/h},~n=1,2,3...,$ at which $J_{s}=0.$ In a vicinity of these points there are the valleys of *mode (1b)* with the width, $\Delta d_{Fn}\approx \xi _{F}\Gamma \Gamma _{BI}^{-1}h^{-1/2}\exp
\{\pi (n-3/4)\},$ embedded into the areas occupied by *mode (1a)*. For the set of parameters used for calculation of the phase diagram presented in Fig.\[Phase\], there is only one valley with the width $\Delta
d_{F1}\approx \xi _{F}\Gamma \Gamma _{BI}^{-1}h^{-1/2}\exp \{\pi /4\}$ located around the point $d_{F1}=(\pi /4)\xi _{F}\sqrt{2/h}$ of the first $0$ to $\pi $ transition.

### Mode (1a): Switchable $0-\protect\pi $  SIs junction

In the experimentally realized case [@Bolginov; @Ryazanov3; @Larkin; @Vernik] $\Gamma _{BI}^{-1}\ll |\beta |\Gamma ^{-1}$ the condition is fulfilled and the weak place in SIsFS structure is located at the SIs interface. In this approximation it follows from (\[EqForKhi1\]) that $$\chi \approx \varphi -\frac{2q\Gamma }{(p-q)\Gamma _{BI}}\sin \left( \varphi
\right)$$in 0-state ($d_{F}<d_{F1}$) 