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

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

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

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