I am trying to find a solution to not display the name and the symbol on the Legend on my ArcMap Layout but I do want to keep the symbol in the Layout .
Here is the screenshot
I have ArcGIS Desktop 10.5.1 here
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One workaround I know to accomplish this is to make a duplicate data frame where you remove that value from being displayed in the layer properties symbology tab. You can then add the legend from that duplicate data frame to your layout. Just make sure to move the map of the duplicate data frame outside of your layout display area.
It appears that your point layer is symbolized "by category", and you want to remove one of those categories from the legend but keep it visible on the map. What you have to do in this case is add the point layer to your map twice. Symbolize one of those layers as you have it in your screen shot, but without the one category you don't want shown in the legend. Add this point layer to your legend. Then symbolize the second instance of this point layer using only the category you want shown on the map, but not in the legend. Do NOT add this layer to the map legend. You can use a definition query on each of these point layers to filter out the categories of the features you want to show.
If I have understood the question. You have a point file called: "Populated Places", it contains information you feel is not important to be included in the Legend of the map.
The easiest way to include data in the Layout View / Map is to simply remove it from the "Legend Properties".
To do this:
Map Layers - contains everything in the ToC
Legend Items - contains all that will appear in the Legend;
That layer will be removed from the Legend.
You can also play around with the "Map Connection" options to allow for layers which are unticked in the ToC to be displayed in the Legend, and locks the layers in place in the Legend if you move them around the ToC.