You may consider the paper:
 
B. Ibrahimpasic, A parametric family of quartic Thue inequalities. 
Bull. Malays. Math. Sci. Soc. (2) 34 (2011), no. 2, 215–230, 
available at [http://www.emis.de/journals/BMMSS/vol34_2_2.html][1] 

It seems that Theorem 3.1, with c=2k, answers your question.


  [1]: http://www.emis.de/journals/BMMSS/vol34_2_2.html