Is there one or more solutions to this system of stiff linear equations?

it is a question of hydrodynamics,and i have evaluated all the known Physical quantity,so there are only 6 unknown,

I use the default of the solve function, but it can’t get a solution,

sol = Solve[{alpha[0] ==  6.9452661266553415` - (2.691307288165229`*^61 -      6.1630057357396486`*^60 I) aa[0] +   0.0005756591988055512` aa[1] + 1.4836217721967435`*^-8 aa[2],  alpha[1] ==  6.9452661266553415` + (2.690177977346087`*^61 -      6.1604196508706296`*^60 I) aa[0] -   0.0005795167180894829` aa[1] - 1.4847619644675353`*^-8 aa[2],  alpha[2] ==  6.9452661266553415` - (2.6867957242378484`*^61 -      6.152674401787605`*^60 I) aa[0] +   0.0005914058327383139` aa[1] + 1.488193080699163`*^-8 aa[2],  aa[0] == (-0.01865295829650781` +     0.0947158131902396` I) (-7.853457866365679`*^61 -     5.3448365305565616`*^60 alpha[1] +     4.105427712466962`*^61 alpha[2]),  aa[1] == -26.813731172513215` (-0.03198613745886671` -     0.0021923914657948863` alpha[1] +     0.0172071159126169` alpha[2]),  aa[2] == -15804.107513690864` (-0.0039269646434191684` -     0.00026757553967274273` alpha[1] +     0.0020626199050551887` alpha[2])}, {alpha[0], alpha[1],  alpha[2], aa[0], aa[1], aa[2]}] 

hints are as this,

RowReduce::luc: Result for RowReduce of badly conditioned matrix {{-1.48362*10^-8+0.     I,-0.000575659+0. I,7.39164*10^62+1.07904*10^62 I,0. +0. I,0. +0. I,1. +0. I,-    6.94527+0. I},{1.48476*10^-8+0. I,0.000579517 +0. I,-7.38868*10^62-1.0786*10^62 I,0.     +0. I,1. +0. I,0. +0. I,-6.94527+0. I},<<3>>,{1. +0. I,0. +0. I,0. +0. I,32.5979 +0.     I,-4.22879+0. I,0. +0. I,-62.0622+0. I}} may contain significant numerical errors.   Solve::svars: Equations may not give solutions for all "solve" variables. 

thanks a lot!