AUSTRIAN-POLISH 1995 Problem 7

Consider the equation $3y^4 + 4cy^3 + 2xy + 48 = 0$ , where $c$ is an integer parameter. Determine all values of $c$ for which the number of integral solutions $(x,y)$ satisfying the conditions (i) and (ii) is maximal: (i) $|x|$ is a square of an integer; (ii) $y$ is a squarefree number.