Let $U$ be an open neighbourhood of $w$ in $\mathbin{\mathbb{R}}$ and $u^{\prime},v^{\prime}:U\rightarrow\mathbin{\mathbb{R}}$ be real analytic functions. If either $a\neq 0$, or $a=0$ and $u^{\prime}(w)\neq 0$, then in an open neighbourhood $V$ of $(w,0)$ in $\mathbin{\mathbb{R}}^{2}$ there exist unique real analytic solutions $u,v:V\rightarrow\mathbin{\mathbb{R}}$ of (33) such that $u(x,0)=u^{\prime}(x)$ and $v(x,0)=v^{\prime}(x)$ for all $x\in U$ with $(x,0)\in V$.