Set your code-viewing glasses to a full 80 columns or more. (Find it in your user settings, if you see red plus marks.) This code tested on 5.8.0 on a variety of Linux architectures.

CAUTION! This program can use a lot of computing resources for very long periods of time. It may take an hour or more just to find the first of thousands of possible solutions.

$ perl quine

For more output, try adding the --verbose command line option. This will cause the program to print incomplete results every few seconds, to help you see what it is doing. These partial results are usually full quines, but this isn't guaranteed. Full solutions are always quines.

#!/usr/bin/perl
$_=$w=q#
$w=~s=\s|\n==sg ;++$L{'                 FfFffFF fFf'};($b,$B,$b b,$BB)
+=(10,6,8,
6);$q=int($b/2) ;($*,$"                 ,$/,$:, $;)=split//,"\0 40:\n\
+043_";$D=
$"x($b+2);$D=$D .($".($                 *x$b).$ ")x$B.$D;$Z=sub {rever
+se@_};++$
L{'IIIIi'};$T=s ub{my($                 x,$P,$T ,$O,$F)=(0,$b*$ q+$q,@
+_);$_=$"x
($b+$b*$b+$b);m y@T=(-2                 -$b,1,2 +$b,-1);@T=&$Z( @{T})i
+f$F;while
($T){$X=substr( $T,0,1,                 '');$x= lc${X}i         f!$x;s
+u
bstr($_,$P,1,$x );(${X}ne$x)?($         P+=$T[$ O]):($O =($O+1)%(@T))}
+s =\A$"+=
=s;s=$"+\z==s;$ _};++$L{'LlLLLl         '};$W=s ub{$_=s hift;eval(qq%y
+/ $"f-z/.
$*/%);s-(\.{4,} )-'.{'.length($         1).'}'- ge;qr/$ _/};++$L{'NNnn
+n NnNn'};
++$L{'PPpPpPp'} ;$r=sub{local($         _,$m)=@ _;s/$"/ /g;s/(.)/$1x$b
+b /ge;$x=
$bb*$b;s|(.{$x} )|$1x$BB|eg;s~(         ?<=(.)) (.)~($1 eq$2)?$2:$*~eg
+; $x--;s~
(?<=(.)         .{$x})(                 .)~($2n e$1)?$* :$2~ge;       
+  s|.(.{$
x})|\1$ /|g;s!.*?$/!!s;                 $x=$w.$ ;.$w;s: \S:subs tr($x,
+0,1,''):g
e;print "$:!$^X$/",q@$_                 =$w=q@, $:,$/,$ _,"$:", q@=>s=
+\s|\n|@,$
;,$;,q@ .*==sg=>eval@,$                 /,"$;$; DATA$;$ ;$*$m$/ ";};++
+$L{'TTtTt
tTTt'}; $L=sub{my($Z,$O                 ,$F);fo r$x(key s%L){$L {$x}={
+};for$O(0
..3){fo r$F(0,1){$Z=&$T                 ($x,$O, $F);$L{ $x}{$Z} =&$W($
+Z);}}}};+
                                        +$L{'Uu UUuUu'}         ;($L5,
+$
L4,$L3,$LL)=($b -4,$b-3         ,$b-2,$ b-1);($ S,$s)=(         "S-S$L
+L :","S--
S$LL:");$h=["S$ b:S-S$b         :S","S$ b:${S}S -S$b:S"         ,"SS$L
+L :${s}SS
","S$b:${S}${S} S-S$b:S         ","SSS$ L4:S--- S$L4:SS         ","S$b
+: ${S}${s
}SS","S$b:S-S$L 3:${s}S         S","SS$ LL:S--S $L3:S-S         $b:S",
+" SS$LL:$
{s}S-S$b:S","S$ b:$S$S$         {S}S-S$ b:S","S SS$L5:S         ----S$
+L 5:SS","
SS$LL:S         --S$L3:         ${s}SS" ,"S$b:$                 S$S$S$
+S ${S}S-S
$b:S"]; for(@$h){s=S=\134S=g;s= \-=$*=g ;s=(\d+                 ):=.{$
+1 }=g;$_=
qr/$_/} ++$L{'VVvVVv'};$H=sub{m y$X=shi ft;for$                 x(@$h)
+{ return'
'if$X=~ /$x/}1;};++$L{'WwWwwwWw Ww'};$V =sub{my                 ($I,$D
+, $q,$Q)=
(0,@_); for$I(0..-1+length($Q)) {$_=sub str($Q,                 $I,1);
+i f($"eq$
_){next }substr($D,$I+$q,1,$_); }&$H($D )&&&$A(                 $D);};
++ +$L{'XX
xxXxXxx         XXx'};$         S=$c=0;                               
+  ++$L{'Y
yYyyYYYy'};$A=s ub{++$c ;my($D)=@_;!@A&&(&$r($D         ,"SOLUTION".++
+$S),retur
n);@ARGV&&(@A<3 )&&&$r( $D,"$c$*moves");my($aa,         $a)=('',pop@A)
+;for$aa(s
ort(keys%{$L{$a }})){wh ile($D=~m/(?=$L{$a}{$aa         })/g){&$V($D,p
+os($D),$a
a);++pos($D);}} push@A, $a;};++$L{'ZzZZzzzZz'};         &$L;@A=&$Z(sor
+t(keys%L)
);&$A($D);__@@* %&(*%(* @&%(*&@^_$w=~s=\s|\n==s         g;++$L{'FfFffF
+FfFf'};($
#=>s=\s|\n|__.*==sg=>eval
__DATA__                                            [ e d @ h a l l e 
+y . c c ]
[download]

Watch this space. I'll insert a link to my POD documentation which gives the blow-by-blow explanation in a few days. For now, the name and the visuals are all I'll give away.