The enzyme deficient in Lesch Nyhan syndrome is:
Ref. Guyton. 13th edition. Page. 879
Lesch-Nyhan syndrome results from a defect in HGPRT, an enzyme involved in the purine salvage pathway.

HGPRT catalyzes the conversion of the free base (hypoxanthine of guanine) to a nucleotide (IMP or GMP).
HGPRT is not required for de novo purine synthesis, or urate synthesis, or for either the de novo or salvage pathways of pyrimidine metabolism.  

Features

Spastic cerebral palsy 


Self-mutilation (hands, lips) 
Hyperuricemia and gout
Early death
 X-linked (recessive)
Adenosine Deaminase
PRPP synthetase
HGPRTase
Xanthine oxidase