My mother applied visit visa in March and refused. I sent PAP and application was reconsidered and interview was conducted. Now ECO said that you can visit each other some other country. I am refugee and can't go back to Pakistan. There is no other issue regarding Provenance and Purpose of visit is raised. I need information and advice whether they legally can refuse visit visa application saying this. Please guide me and suggest me any case law or authority relevant to it.
Immigration authorities are legally able to refuse visas for almost any reason. If you think there's an error here then usually your only recourse is to apply again, or perhaps apply for a judicial review (if that's appropriate for the jurisdiction to which you're applying).
You've tried the first route, and the second is prohibitively expensive with no guarantee of success
The ECO has suggested you meet in some third country. Visa Guide has a list of visa-free and visa-on-arrival countries available to Pakistanis. Perhaps there's somewhere there you could both go to?
You say that ‘purpose of visit was raised’. If you mean that this was given as a refusal reason, the key problem here could be doubt about whether she genuinely intends to visit, in the light of her circumstances including your own immigration status and her home country of Pakistan.
Immigration officials consider several factors when assessing immigration risk, including family and economic ties to the applicant’s country of residence and whether they have family members in the jurisdiction to which they are applying, and the political, economic and security situation in the applicant’s country of residence. Based on the limited information you provided, your mother’s overall circumstances could well have lead to doubts about her intention to leave, and thence to a reasonable conclusion that her real intention may be to overstay or even apply for asylum. As Redd Herring’s answer says, your only recourse is to seek help from an Immigration practitioner, but in the absence of an error on the part of the visa-issuing authority this is unlikely to succeed.