How to keep already-set GET parameter values on form submission?

You can add two hidden fields in the form on the first target site, blabla.php in your case:

<form ...>
  <input type="hidden" name="name" value="<?php echo htmlspecialchars($_GET['name']);?>">
  <input type="hidden" name="lName" value="<?php echo htmlspecialchars($_GET['lName']);?>">

  <!-- rest of the form here -->
</form>

For a dynamic solution, use a foreach loop:

<?php
foreach($_GET as $name => $value) {
  $name = htmlspecialchars($name);
  $value = htmlspecialchars($value);
  echo '<input type="hidden" name="'. $name .'" value="'. $value .'">';
}
?>

You may consider locking the dynamic approach down to a list of known possible keys:

<?php
$keys = array('name', 'lName', ...);
foreach($keys as $name) {
  if(!isset($_GET[$name])) {
    continue;
  }
  $value = htmlspecialchars($_GET[$name]);
  $name = htmlspecialchars($name);
  echo '<input type="hidden" name="'. $name .'" value="'. $value .'">';
}
?>

A simpler solution to keep the URL unchanged by using http_build_query

 <form action="<?php echo $_SERVER["PHP_SELF"] . '?'.http_build_query($_GET); ?>" ... 
  ..
  ..