We give a systematic discussion of the relation between q-difference equations which are conditionally $U_q\left(\right)$-invariant and subsingular vectors of Verma modules over $U_q\left(\right)$ (the Drinfeld-Jimbo q-deformation of a semisimple Lie algebra over ℂg or ℝ). We treat in detail the cases of the conformal algebra, = su(2,2), and its complexification, = sl(4). The conditionally invariant equations are the q-deformed d’Alembert equation and a new equation arising from a subsingular vector proposed by Bernstein-Gel’fand-Gel’fand....