Please let me know how this round sink is mounted. There's no lip and I don't know how the brackets would go on it. Thanks.
Looking at the top back of the sink, there are two screw holes provided, but these are for location not full weight bearing.
Having removed a similar sink in the past, there should be two (ornate) cast iron arms which may also double as towel rails which are secured to the wall and support the weight of the sink.
Another version would have two posts that came up from the floor to the side/front of the sink, there may be evidence of fixing points if you look - the picture does not show us. You might consider an under-sink cupboard with short support posts...
If you don't have those arms or posts then you will need to make something equivalent as you will not be able to rely on those two screw holes.
You might find some searching on google etc will provide images of similar sinks with the supports shown.
Not sure what the model and make is, but this appears to be a) old and b) heavy iron or steel. Furthermore the recess is beveled in such a way as to dislike any type of weight bearing. If it's steel you definitely NEED a mounting bracket.
Two options are as follows:
cut a piece of wood and prime it heavily, that fits in the recess, and then against the wall side, and on the upper and lower part, weld on a lip which holds the wood in place. (If done carefully it will not damage any porcelain or enamel on the front side) Bolt the bracket to the wood then. You'd need to route out the wood to accommodate the metal lips.
skip the wood entirely and just weld a metal plate to the back and mount the mounting plate via bolts.
In either case you need to make sure that the lip of the mounting plate "sucks the sink" up against the wall and is not loose, as this tightness transfers the load to whatever you do.