  We investigate chaotic to integrable transition in two types of hybrid SYK
models which contain both $ q=4 $ SYK with interaction $ J $ and $ q=2 $ SYK
with an interaction $ K $ in type-I or $(q=2)^2$ SYK with an interaction $
\sqrt{K} $ in type-II. These models include hybrid Majorana fermion, complex
fermion and bosonic SYK. For the Majorana fermion case, we discuss both $ N $
even and $ N $ odd case. We make exact symmetry analysis on the possible
symmetry class of both types of hybrid SYK in the 10 fold way by Random Matrix
Theory (RMT) and also work out the degeneracy of each energy levels. We
introduce a new universal ratio which is the ratio of the next nearest
neighbour (NNN) energy level spacing to characterize the RMT. We perform exact
diagonalization to evaluate both the known NN ratio and the new NNN ratio, then
use both ratios to study Chaotic to Integrable transitions (CIT) in both types
of hybrid SYK models. Some preliminary results on possible quantum analog of
Kolmogorov-Arnold-Moser (KAM) theorem and its dual version in the quantum
chaotic side are given. We explore some intrinsic connections between the two
complementary approaches to quantum chaos: the RMT and the Lyapunov exponent by
the $ 1/N $ expansion in the large $ N $ limit at a suitable temperature range.
Comments on some previously related works are given. Some future perspectives,
especially the failure of the Zamoloddchikov's c-theorem in 1d CFT RG flow are
outlined.
