Let $D$ be a $2$ dimensional distribution of $\mathbb{R}^3$. Is there a $1$ dimensional foliation of $\mathbb{R}^3$ with Frennet curves such that for every leaf $\gamma$ of the foliation we have $Span(\gamma'(t), \gamma ''(t))=D(\gamma(t))$ where $\gamma(t)$ is the unit speed parametrization of the leaf $\gamma$?

One can generalize the question when $\mathbb{R}^3$ is equiped with an arbitrary Riemannian metric and we require a foliation with $span(\gamma'(t), \nabla_{\gamma'(t)} \gamma '(t))=D(\gamma(t))$.

Let $D$ be a $2$ dimensional distribution of $\mathbb{R}^3$. Is there a $1$ dimensional foliation of $\mathbb{R}^3$ with Frenet curves such that for every leaf $\gamma$ of the foliation we have $\mathrm{span}(\gamma'(t), \gamma ''(t))=D(\gamma(t))$ where $\gamma(t)$ is the unit speed parametrization of the leaf $\gamma$?

One can generalize the question when $\mathbb{R}^3$ is equiped with an arbitrary Riemannian metric and we require a foliation with $\mathrm{span}(\gamma'(t), \nabla_{\gamma'(t)} \gamma '(t))=D(\gamma(t))$.