Consider the Riemann-Hilbert boundary value problems for elliptic systems in the plane (1) \[ DW+\epsilon \sum^{r-1}_{k=0}e^ k\sum^{r- 1}_{i=0}(A_{k_ j}W_ j+B_{k_ j}\bar W_ j)=h,\quad z\in \Omega \]
\[ Re[\lambda (z)W(z)]=g(z),\quad z\in \Gamma \]
\[ D=\partial /\partial x+i\partial /\partial y+e(a\partial /\partial x+b\partial /\partial y) \] where \(\epsilon\) is a parameter, \(A_{k_ j},B_{k_ j}\in L_ p(\Omega)\), \(p>2\); \(h(z)\in L_ p(\Omega)\), \(\lambda (z)\in H_{\alpha}(\Gamma)\) and \(\lambda_ 0(z)\neq 0\); \(\Omega\) is a unit disk; \(\Gamma\) : \(| z| =1\). The solution of (1) is constructed by using a class of integral representation formulas and the problem is transformed into a Fredholm integral equation of second kind.