Isospin symmetry in mirror $\alpha$-decays

We show that a consequence of isospin symmetry, recently discovered in mirror conjugated one-nucleon decays, can be extended to mirror-conjugated $\alpha$-particles decays, both virtual and real. For virtual $\alpha$-decays of bound mirror pairs this symmetry manifests itself as a relation between the Asymptotic Normalization Coefficients (ANCs) of $\alpha$-particle overlap integrals. This relation is given by a simple analytical formula which involves $\alpha$-particle separation energies and charges of residual nuclei. For bound-unbound mirror pairs, the ANC of a bound nucleus is related to the $\alpha$-width of the mirror unbound level. For unbound mirror pairs we get a new analytical formula that relates the widths of mirror resonances. We test the validity of these analytical formuli against the predictions of a two-body potential and of a many-body microscopic cluster model for several mirror states in $^7$Li-$^7$Be, $^{11}$B-$^{11}$C and $^{19}$F-$^{19}$Ne isotopes. We show that these analytical formulae are valid in many cases but that some deviations can be expected for isotopes with strongly deformed and easily excited cores. In general, the results from microscopic model are not very sensitive to model assumptions and can be used to predict unknown astrophysically relevant cross sections using known information about mirror systems