This trial is testing if giving synthetic growth hormone to veterans with Gulf War Illness and growth hormone deficiency can reduce body fat and improve their symptoms. The study will check if this treatment is safe and effective over several months. If successful, it could lead to better care for these veterans. Growth hormone (GH) treatment has been shown to increase height velocity in children with Prader-Willi syndrome, decrease weight-for-height index values and body fat mass, and have a positive effect on lean body mass during therapy.