= Solution
Let $J$ be rotation by ninety degrees in the oriented tangent plane, $Jv=N\times v$. In local coordinates seek a curve with
$$
\nabla_{\dot\gamma}\dot\gamma=\lambda J\dot\gamma,
\quad\text{or}\quad
\ddot x^k+\Gamma^k_{ij}\dot x^i\dot x^j=\lambda J^k_i\dot x^i.
$$
This smooth second-order <ordinary differential equation>, written as a first-order system for $(x,\dot x)$, has a unique local solution with the specified initial position and velocity. Its speed is constant, since
$$
\frac d{dt}|\dot\gamma|^2=2\langle\nabla_{\dot\gamma}\dot\gamma,\dot\gamma\rangle
=2\lambda\langle J\dot\gamma,\dot\gamma\rangle=0.
$$
The initial speed is one, so the parameter is arc-length and the signed <geodesic curvature> is $\langle\nabla_s\dot\gamma,J\dot\gamma\rangle=\lambda$. Conversely every requested curve satisfies this system, which proves the claimed uniqueness as well as existence on some $(-\epsilon,\epsilon)$.
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