External or internal companion exciting the spiral arms in CQ Tau?

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.