  We consider $\beta$ matrix models with real analytic potentials. Assuming
that the corresponding equilibrium density $\rho$ has a one-interval support
(without loss of generality $\sigma=[-2,2]$), we study the transformation of
the correlation functions after the change of variables
  $\lambda_i\to\zeta(\lambda_i)$ with $\zeta(\lambda)$ chosen from the equation
$\zeta'(\lambda)\rho(\zeta(\lambda))=\rho_{sc}(\lambda)$, where
$\rho_{sc}(\lambda)$ is the standard semicircle density. This gives us the
"deformed" $\beta$-model which has an additional "interaction" term. Standard
transformation with the Gaussian integral allows us to show that the "deformed"
$\beta$-model may be reduced to the standard Gaussian $\beta$-model with a
small perturbation $n^{-1}h(\lambda)$. This reduces most of the problems of
local and global regimes for $\beta$-models to the corresponding problems for
the Gaussian $\beta$-model with a small perturbation. In the present paper we
prove the bulk universality of local eigenvalue statistics for both one-cut and
multi-cut cases.
