and I do not quite understand you mean by "低收入仍有资格由律师转介服务给有愿意的律师

I do not quite understand what did you mean by "低收入仍有资格由律师转介服务给有愿意的律师接手." One is criminal, the other is civil. In a criminal case, she has the constitutional right to a court appointed lawyer if she cannot afford one. But in a civil case, if she does not want to pay upfront, she has to convince a lawyer that her case is strong enough so he can take it on a contingency basis.