  In the present paper, we prove the existence of solutions
$(\lambda_1,\lambda_2,u,v)\in\mathbb{R}^2\times H^1(\mathbb{R}^3,\mathbb{R}^2)$
to systems of coupled Schr\"odinger equations $$ \begin{cases} -\Delta
u+\lambda_1u=\mu_1 u^3+\beta uv^2\quad &\hbox{in}\;\mathbb{R}^3\\ -\Delta
v+\lambda_2v=\mu_2 v^3+\beta u^2v\quad&\hbox{in}\;\mathbb{R}^3\\
u,v>0&\hbox{in}\;\mathbb{R}^3 \end{cases} $$ satisfying the normalization
constraint $
\displaystyle\int_{\mathbb{R}^3}u^2=a^2\quad\hbox{and}\;\int_{\mathbb{R}^3}v^2=b^2,
$ which appear in binary mixtures of Bose-Einstein condensates or in nonlinear
optics. The parameters $\mu_1,\mu_2,\beta>0$ are prescribed as are the masses
$a,b>0$. The system has been considered mostly in the fixed frequency case. And
when the masses are prescribed, the standard approach to this problem is
variational with $\lambda_1,\lambda_2$ appearing as Lagrange multipliers. Here
we present a new approach based on bifurcation theory and the continuation
method. We obtain the existence of normalized solutions for any given $a,b>0$
for $\beta$ in a large range. We also give a result about the nonexistence of
positive solutions. From which one can see that our existence theorem is almost
the best. Especially, if $\mu_1=\mu_2$ we prove that normalized solutions exist
for all $\beta>0$ and all $a,b>0$.
