  We introduce two disorder-free variants of the Sachdev-Ye-Kitaev (SYK) model,
demonstrate their integrability, and study their static and dynamical
properties. Unlike diagrammatic techniques, the integrability of these models
allows us to obtain dynamical correlation functions even when the number of
Majorana fermions is finite. From the solutions, we find that out-of-time-order
correlators (OTOCs) in these models exhibit exponential growth at early times,
resembling that of quantum chaotic systems such as those with disorder or
external kick terms. Conversely, our analysis shows no evidence of
random-matrix behavior in level statistics or the spectral form factor. Our
findings illustrate that the clean versions of the SYK models represent simple
but nontrivial examples of disorder-free quantum many-body systems displaying
chaos-like behavior of OTOCs.
