The $^{22}$Ne($alpha$,$gamma$)$^{26}$Mg and $^{22}$Ne($alpha$,n)$^{25}$Mg reactions play an important role in astrophysics because they have significant influence on the neutron flux during the weak branch of the s-process. We constrain the astrophysical rates for these reactions by measuring partial $alpha$-widths of resonances in $^{26}$Mg located in the Gamow window for the $^{22}$Ne+$alpha$ capture. These resonances were populated using $^{22}$Ne($^6$Li,d)$^{26}$Mg and $^{22}$Ne($^7$Li,t)$^{26}$Mg reactions at energies near the Coulomb barrier. At these low energies $alpha$-transfer reactions favor population of low spin states and the extracted partial $alpha$-widths for the observed resonances exhibit only minor dependence on the model parameters. The astrophysical rates for both the $^{22}$Ne($alpha$,$gamma$)$^{26}$Mg and the $^{22}$Ne($alpha$,n)$^{25}$Mg reactions are shown to be significantly different than the previously suggested values.
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