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The main focus of this paper is the following matrix Cauchy problem for theDirac system on the interval $[0,1]:$ \[ D'(x)+\left[\begin{array}{cc} 0 &\sigma_1(x)\\ \sigma_2(x) & 0 \end{array} \right]D(x)=i\mu\left[\begin{array}{cc} 1 & 0\\ 0 &-1 \end{array} \right]D(x),\quadD(0)=\left[\begin{array}{cc} 1 & 0\\ 0 & 1 \end{array}\right], \] where$\mu\in\mathbb{C}$ is a spectral parameter, and $\sigma_j\in L_2[0,1]$,$j=1,2$. We propose a new approach for the study of asymptotic behaviour of itssolutions as $\mu\to \infty$ and $|{\rm Im}\,\mu|\le d$. As an application, weobtain new, sharp asymptotic formulas for eigenfunctions of Sturm-Liouvilleoperators with singular potentials.

On asymptotic behaviour of solutions of the Dirac system and applications to the Sturm-Liouville problem with a singular potential