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                                                                     EXHIBIT 3.4



                                 FIRST AMENDMENT

                                       TO

                    RIGHTS AGREEMENT DATED FEBRUARY 26, 1999

         This First Amendment dated February 9, 2001 (this "Amendment"), between
UTI Energy Corp., a Delaware corporation (the "Company"), and Mellon Investor
Services LLC, a New Jersey limited liability company formerly known as
ChaseMellon Shareholder Services, L.L.C. (the "Rights Agent"), amending the
Rights Agreement dated February 26, 1999 (the "Rights Agreement"), between the
Company and the Rights Agent.

         WHEREAS, the Company has entered into our Agreement and Plan of Merger
dated as of February 4, 2001 (the "Merger Agreement), pursuant to which the
Company will merge (the "Merger") into Patterson Energy, Inc., a Delaware
corporation ("PEC");

         WHEREAS, the Board of Directors of the Company has approved the Merger
Agreement and the Merger and has determined to amend the Rights Agreement so
that (i) PEC and its Affiliates and Associates (as defined in the Rights
Agreement) shall not be deemed to by an Acquiring Person (as defined in the
Rights Agreement) and that no Distribution Date (as defined in the Rights
Agreement) shall occur as a result of the execution and delivery of the Merger
Agreement or the consummation of the Merger as contemplated thereby; and

         WHEREAS, the Board of Directors of the Company has determined to amend
the Rights Agreement so that Section 13 thereof shall not apply to the Merger as
contemplated by the Merger Agreement.

         NOW, THEREFORE, it is agreed as follows:

         1. Notwithstanding anything in the Rights Agreement to the contrary,
(a) PEC and its Affiliates and Associates shall not be deemed to be an Acquiring
Person on account of the execution and delivery of the Merger Agreement or the
announcement thereof or as a result of or arising out of the Merger; (b) no
Distribution Date shall have occurred, shall occur or shall be deemed to occur
as a result of the execution and delivery of the Merger Agreement or the
announcement thereof or as a result of or arising out of the Merger; and (c)
Sections 13 and 25(a)(iv) of the Rights Agreement shall not apply to the Merger.

         2. Except as amended or modified pursuant to this Amendment, the Rights
Plan remains in full force and effect and this Amendment shall be subject to the
terms thereof except or expressly modified hereby.

         Executed as of the date first written above.



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                                        UTI ENERGY CORP.



                                        By    /s/ John E. Vollmer III
                                            ------------------------------------
                                                 John E. Vollmer III
                                                 Senior Vice President
                                                 and Chief Financial Officer


                                        MELLON INVESTOR SERVICES
                                        LLC, as Rights Agent



                                        By   /s/ Deodatt Lakeram
                                            ------------------------------------
                                                 Name: Deodatt Lakeram
                                                 Title: Assistant Vice President



                                   CERTIFICATE


         The undersigned, John E. Vollmer, Senior Vice President and Chief
Financial Officer of UTI Energy Corp. ("UTI") hereby certifies that the First
Amendment dated February 9, 2001, to which this Certificate is attached, to the
Rights Agreement dated February 26, 1999, between UTI and MELLON INVESTOR
SERVICES LLC is in compliance with the provisions of Section 27 of such Rights
Agreement and does not increase the Rights Agent's duties, liabilities or
obligation under such Rights Agreement.

         Signed this 9th day of February, 2001.



                                           /s/ John E. Vollmer III
                                        ----------------------------------------
                                        John E. Vollmer III
                                        Senior Vice President
                                        and Chief Financial Officer