We investigate the equilibrium dynamics of our recently proposed toy model of dense fluid in the infinite damping limit. Contrary to naive expectation, the correlators involving the velocity-like variables do not quickly relax away. Instead, after a very fast transient relaxation, they exhibit rather slow relaxations due to the coupling to the density-like variable. Hence, the so-called "hopping" processes are not suppressed even in the large damping limit. These hopping processes can only be controlled by tuning the parameter delta* which is the ratio of the numbers of the components of the velocity-like and the density-like variables in the model. We analytically prove that there must exist an ergodic-to-nonergodic phase transition for delta* such that 0 < delta* < 1. The slow dynamics and the dynamic transition in the model are distinct from those in the idealized mode coupling theory.