renzh: since your matrix is singular, there is not a unique solution. Here the following solvers give me a solution with (A*x-b).cwiseAbs().maxCoeff() around 1e-11:

MatrixXcd A;
VectorXcd x, b;
SparseMatrix<double> S;
loadMarket(S,"Krr_1.dat");
int n = S.cols();
A = MatrixXcd(S);
x.setOnes(n);
b = A*x;

x.setZero();
x = SimplicialLDLT<SparseMatrix<double> >(S).solve(b);
//x = A.ldlt().solve(b);
//x = A.partialPivLu().solve(b);
//x = A.colPivHouseholderQr().solve(b);
std::cout << x.transpose() << "\n";
std::cout << (A*x-b).cwiseAbs().maxCoeff() << "\n";