...
$form['key'] = array(
    '#type' => 'fieldset',
    '#title' => t('Key'),
    '#collapsible' => TRUE,
    '#collapsed' => FALSE,
  );
  $form['key']['first'] = array(
    '#type' => 'textfield',
    '#title' => t('Введите ключевые слова поиска'),
  //  '#default_value' => $form_state['values']['first'], // changed  // replaced
    '#description' => "Please enter keys.",
    '#size' => 20,
    '#maxlength' => 20,
  );

...

  $form['next'] = array(
    '#type' => 'submit',
    '#value' => 'Done',
  );
...