Let $X$ be a nowhere-zero $C^{\infty}$ complex vector field defined near the origin in $\Bbb R^2$. We may suppose that $X$ has the form of $\frac{\partial}{\partial t}+ir(t,x)\frac{\partial}{\partial x}$, where $r(t,x)$ is a real-valued $C^{\infty}$ function. Up to now the investigations on local integrability for vector fields $X$ satisfying $r(0,x)=0$ have been focused. This paper treats the vector field $X$ of the form of $$ \frac{\partial}{\partial t}+i\bigl(t^d+a(x)\bigr)\frac{\partial}{\partial x}, $$ where $d$ is a positive integer and $a(x)$ a real-valued $C^{\infty}$ function satisfying $a(0)=0$. Under certain assumptions on $a(x)$, the following properties are shown: \proclaim { Property A}Every $C^1$ solution $u$ of the equation $Xu=0$ in a neighborhood of the origin which satisfies that $u(0,x)$ is constant is identically constant. \endproclaim \proclaim { Property B}Every $C^2$ solution $u$ of the equation $Xu=0$ in a neighborhood of the origin which satisfies $u(t,x)-u(-t,x)=o(t^2) (t\to0)$ is identically constant. \endproclaim