Reverse and add degeneracy
Ruby, 84 79 77 74 bytes



-&gt;x{y=1;x+="#{x}".reverse.to_i while(2...x).any?{|z|0==y=x%z}&amp;&amp;x%11&gt;0;y&gt;0}


Try it online!


   If I got it right, when we reach a multiple of 11 we can stop (we will only get multiples of 11 after that)