the lattice $16^4$. The transition point at $\lambda = 0.009$ was checked using the larger lattice ($20^3\times 24$). In order to simulate the system we used Metropolis algorithm. The acceptance rate is kept around $0.5$ via the automatical self - tuning of the suggested distribution of the fields. At each step of the suggestion the random value is added to the old value of the scalar field while the old value of Gauge field is multiplied by random $SU(2)\otimes U(1)$ matrix. We use Gaussian distribution both for the random value added to the scalar field and the parameters of the random matrix multiplied by the lattice Gauge field. We use two independent parameters for these distributions: one for the Gauge fields and another for the scalar field. The program code has been tested for the case of frozen scalar field. And the results of the papers [@VZ2008] are reproduced. We also have tested our code for the $U(1)$ field frozen and repeat the results of [@Montvayold]. Far from the transition point the autocorrelation time for the gauge fields is estimated as about $N^g_{auto} \sim 500$ Metropolis steps. In the vicinity of the transition point the autocorrelation time is several times 