My question is somehow related to: The relation between critical surface and the (renormalization) fixed point but there is another problem:
The problem is that if we accept that all points on the critical surface are critical in the manner that their corresponding correlation length is infinite, then according to the scaling hypothesis a system whose parameters lie on the critical surface should be scale invariant. And therefore its parameters shouldn't change under the RG transformation. So each point on the critical surface should be a fixed point and hence there is not any RG flow over the critical surface. It implies that the RG transformation doesn't push any point on the critical surface to our first fixed point.
From the above statements one can deduce that the RG method is inconsistent. Surely there should be something which I'm missing but what's that?