  We present new high-contrast images in near-infrared wavelengths ($\lambda$ =
1.04, 1.24, 1.62, 2.18 and 3.78$\mu$m) of the young variable star CQ Tau,
aiming to constrain the presence of companions in the protoplanetary disc. We
reached a Ks-band contrast of 14 magnitudes with SPHERE/IRDIS at separations
greater than 0."4 from the star. Our mass sensitivity curve rules out giant
planets above 4 M$_{\rm Jup}$ immediately outside the spiral arms at $\sim$60
au and above 2-3 M$_{\rm Jup}$ beyond 100 au to 5$\sigma$ confidence assuming
hot-start models. We do, however, detect four spiral arms, a double-arc and
evidence for shadows in scattered light cast by a misaligned inner disc. Our
observations may be explained by an unseen close-in companion on an inclined
and eccentric orbit. Such a hypothesis would also account for the disc CO
cavity and disturbed kinematics.
