Question

If solve is user, I need the value of name (LINES).

if solve is imm and name is ARCH, I need the value of value (6M)

I tried in the following:

use strict;
use warnings;
use XML::LibXML;
my $parser = XML::LibXML->new();
my $doc = $parser->parse_file(...);
my $j = 0;
my @imm;
$imm[$j++] = $_->data
    foreach 
        $doc->findnodes(
            "//meters/meter"
            ."/name[../value[\@solve/string() = 'immediate']]"
            ."/text()");

Errors are

Possible unintended interpolation of @solve in string
Global symbol "@solve" requires explicit package name 

Sample XML:

<rules> 
      <meters>        
             <meter>
                   <name>LINES</spirit:name>
                   <display>LINES</display>
                   <description>Specifies the number of lines</description>
                   <value
                         solve="user"
                         id="LINES"
                         order="11"
                         format="long"
                         dataType="integer" 
                         min="2" 
                         max="34">34</value>
            </meter>
            <meter>
                  <name>ARCH</name>
                  <display>Define ARCH</display>
                  <description>Define ARCH</description>
                  <value 
                        format="string" 
                        solve="imm" 
                        id="ARCH" 
                        >6M</value>
             </meter>
       </meters>    
</rules>

<rules> 
      <meters>        
             <meter>
                   <name>LINES</spirit:name>
                   <display>LINES</display>
                   <description>Specifies the number of lines</description>
                   <value
                         solve="user"
                         id="LINES"
                         order="11"
                         format="long"
                         dataType="integer" 
                         min="2" 
                         max="34">34</value>
            </meter>
            <meter>
                  <name>ARCH</name>
                  <display>Define ARCH</display>
                  <description>Define ARCH</description>
                  <value 
                        format="string" 
                        solve="imm" 
                        id="ARCH" 
                        >6M</value>
             </meter>
       </meters>    
</rules>

Answer 1

This works for me:

#!/usr/bin/perl
use strict;
use warnings;

use XML::LibXML;

my $parser = XML::LibXML->new();
my $doc = $parser->parse_file('XXX.xml');
my $j = 0;
my @imm;
$imm[$j++] = $_->data
    for $doc->findnodes(
            "//meters/meter"
            ."/name[../value[\@solve = 'imm']]"
            ."/text()");
print for @imm;

I made the following changes to your code:

1. I removed the /string() part from the XPath expression. It is invalid usage of a function as indicated by the error I got.
2. The value of solve is imm in your sample data, not immediate.

BTW, changing the middle of the XPath expression to

'/name[../value[@solve = "immediate"]]'

avoids the need of the backslash.