When's the nearest special common year?
PHP, 67 Bytes

for(;date(LN,mktime(0,0,0,1,1,$y=$argn+$i))&gt;1;)$i=($i&lt;1)-$i;echo$y;


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or


for(;date(LN,strtotime("1/1/".$y=$argn+$i))&gt;1;)$i=($i&lt;1)-$i;echo$y;


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Expanded

for(;
date(LN,mktime(0,0,0,1,1,$y=$argn+$i)) # N is 1 for Monday and L is 0 for Non leap year
&gt;1;) # loop so long as expression is not 1
  $i=($i&lt;1)-$i; # set $i 0,1,-1,2,-2 ...
echo$y; # Output Year


date