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/usr/sbin/update-mate-menu-welcome is in ubuntu-mate-welcome 16.04.9.

This file is owned by root:root, with mode 0o755.

The actual contents of the file can be viewed below.

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#!/usr/bin/perl

# This program is free software; you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation; either version 2 of the License, or
# (at your option) any later version.
#
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
# GNU General Public License for more details.
#
# You should have received a copy of the GNU General Public License
# along with this program; if not, write to the
# Free Software Foundation, Inc.,
# 51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA.
#
# Copyright (C) 2007-2015 Oleksandr Shneyder <oleksandr.shneyder@obviously-nice.de>
# Copyright (C) 2007-2015 Heinz-Markus Graesing <heinz-m.graesing@obviously-nice.de>
# Copyright (C) 2013-2015 Mike Gabriel <mike.gabriel@das-netzwerkteam.de>
# Copyright (C) 2015 Martin Wimpress <code@flexion.org>

use strict;

my $mode=shift;
if( $mode eq "--help" )
{
     print "Usage: $0 [--remove]\n";
     print "       Add (or remove) \"Ubuntu MATE Welcome\" button to (from) MATE's Settings menu.\n";
     exit;
}
my $menu;
if ( -e "/etc/xdg/menus/mate-settings.menu" )
{
    $menu="/etc/xdg/menus/mate-settings.menu";
}
else
{
    die "MATE Settings menu not found in /etc/xdg/menus";
}

my $menustr="<MergeFile>/etc/xdg/menus/mate-settings-welcome.menu</MergeFile>\n";

open(FILE,"<$menu");
my @ln=<FILE>;
close(FILE);
my $text=join("",@ln);

if( $mode eq "--remove" )
{
    if($text =~ m/$menustr/)
    {
        $text =~ s/\n$menustr//;
        open(FILE,">$menu");
        print FILE $text;
        close(FILE);
    }
}
else
{
    if(! ($text =~ m/$menustr/))
    {
        $text =~ s/<\/Layout>/<\/Layout>\n$menustr/;
        open(FILE,">$menu");
        print FILE $text;
        close(FILE);
    }
}