I am sorry @Axel Vogt for not being precise.  The physical solution I am interested in has an imaginary part close to + or - 1/2.

I specified the starting values for fsolve and I tried to track down the iterations but it seems that after 7 iterations the solution runs away from the desired solution:

infolevel[fsolve] := 2:

sols := fsolve(Sys2, {u[1] = -.35, u[2] = -.17, u[3] = .17, u[4] = .35, u[5] = 0, v[1] = -.54, v[2] = .50, v[3] = .50, v[4] = -.54, v[5] = -.53});

fsolve/sysnewton: trying multivariate Newton iteration
fsolve/sysnewton:
guess vector Vector(10, [-.35,-.17,.17,.35,0,-.54,.50,.50,-.54,-.53])
fsolve/sysnewton: norm of errors: .5178004668803853520800
fsolve/sysnewton: new norm: 0.3539145219881116705970323967004e-1
fsolve/sysnewton: iter = 1 |incr| = 0.37384e-1 new values u[1] = -.35077 u[2] = -.17006 u[3] = .17006 u[4] = .35077 u[5] = 0.29492e-35 v[1] = -.54860 v[2] = .50837 v[3] = .50837 v[4] = -.54860 v[5] = -.53178
fsolve/sysnewton: new norm: 0.6090108537498864270152480255426e-3
fsolve/sysnewton: iter = 2 |incr| = 0.16994e-1 new values u[1] = -.35214 u[2] = -.17648 u[3] = .17648 u[4] = .35214 u[5] = -0.72215e-36 v[1] = -.54831 v[2] = .50805 v[3] = .50805 v[4] = -.54831 v[5] = -.53157
fsolve/sysnewton: new norm: 0.2318060894787525428080924612269e-6
fsolve/sysnewton: iter = 3 |incr| = 0.28300e-3 new values u[1] = -.35213 u[2] = -.17636 u[3] = .17636 u[4] = .35213 u[5] = -0.36154e-37 v[1] = -.54832 v[2] = .50806 v[3] = .50806 v[4] = -.54832 v[5] = -.53157
fsolve/sysnewton: new norm: 0.2233693226731561926138941452438e-13
fsolve/sysnewton: iter = 4 |incr| = 0.42412e-7 new values u[1] = -.35213 u[2] = -.17636 u[3] = .17636 u[4] = .35213 u[5] = 0.34970e-41 v[1] = -.54832 v[2] = .50806 v[3] = .50806 v[4] = -.54832 v[5] = -.53157
fsolve/sysnewton: new norm: 0.9950011757213036462014345007667e-27
fsolve/sysnewton: iter = 5 |incr| = 0.17376e-13 new values u[1] = -.35213 u[2] = -.17636 u[3] = .17636 u[4] = .35213 u[5] = -0.38025e-42 v[1] = -.54832 v[2] = .50806 v[3] = .50806 v[4] = -.54832 v[5] = -.53157
fsolve/sysnewton: new norm: 0.1413389913634910099563094101840e-29
fsolve/sysnewton: iter = 6 |incr| = 0.62069e-27 new values u[1] = -.35213 u[2] = -.17636 u[3] = .17636 u[4] = .35213 u[5] = -0.19796e-41 v[1] = -.54832 v[2] = .50806 v[3] = .50806 v[4] = -.54832 v[5] = -.53157
fsolve/sysnewton: new norm: 0.1413389913632384303305907431618e-29
fsolve/sysnewton: iter = 7 |incr| = 0.24382e-30 new values u[1] = -.35213 u[2] = -.17636 u[3] = .17636 u[4] = .35213 u[5] = 0.42352e-42 v[1] = -.54832 v[2] = .50806 v[3] = .50806 v[4] = -.54832 v[5] = -.53157
fsolve/sysnewton: new norm: 0.1413389913633003246777836331644e-29
fsolve/sysnewton: new norm: 0.1413389913633003246777836331644e-29
fsolve/sysnewton: new norm: 0.1413389913632384303305907431618e-29
fsolve/sysnewton: new norm: 0.1413389913632384303305907431618e-29
fsolve/sysnewton: new norm: 0.1413389913632384303305907431618e-29
fsolve/sysnewton: new norm: 0.1413389913632384303305907431618e-29
fsolve/sysnewton: new norm: 0.1413389913632384303305907431618e-29
fsolve/sysnewton: new norm: 0.1413389913632384303305907431618e-29
fsolve/sysnewton: new norm: 0.1413389913632384303305907431618e-29
fsolve/sysnewton: new norm: 0.1413389913632384303305907431618e-29
fsolve/sysnewton:
guess vector Vector(10, [-1.372706307028978241251378722551,2.529830542464654567331795848792,-1.360673819312142785520906447408,3.997794043918580166449413416224,-2.269126641933219693171563220696,-6.833450315852802566930712924897,-3.680938534041913165546976837461,.9254988468865938032688258297403,2.910859320164443998796751228316,5.124837060062167853203649854607])
fsolve/sysnewton: norm of errors: 1634634043.879873677131432296702

Thank you  @Axel Vogt  for your feedback.  Surprisingly I do not get the wanted solution. The kind of solution I am interested in has the following form:

{u[1] = -.35212504321763690,

u[2] = -.17636112247467896,

u[3] = .17636112247467896,

u[4] = .35212504321763690,

u[5] = 0,

v[1] = -.54831678079355564,

v[2] = .50806078638906629,

v[3] = .5080607863890662,

v[4] = -.54831678079355564,

v[5] = -.53157245321483338}

the  eval(Sys, test); gives 

{-3.5*10^(-19) = 0, -1.474956135860985261*10^(-19) = 0, -6.27*10^(-18) = 0, -2.98*10^(-18) = 0, 6.64*10^(-18) = 0, 1.3081*10^(-17) = 0, 2.081*10^(-17) = 0, 2.7278*10^(-17) = 0, -1.89663*10^(-16) = 0, 1.027013*10^(-15) = 0}

Do you have some hints to locate the desired solution ?