examples: $q$ is a conjugate point of $p$, but geodesic $pq$ is unique

Question

Let $(M,g)$ be a Riemannian manifold, $p\in M$. It's well known that if the geodesic connecting $p$ to $q$ is not extendable at $q$, then either the geodesic connecting $p$ to $q$ is not unique, or $q$ is a conjugate point of $p$.

Then is there a simple example of $(M,g,p)$ with points conjugate to $p$ but with unique minimal geodesics to $p$?

Answer 1

I think, it is possible when $p$ and $q$ are conjugate along geodesic $\gamma$ that is not minimal for these points.

For example, if $p=(1,0,0)$ and $q=(-1,0,0)\in S^2$, and Riemannean metric on $S^2$ is multiplied by the function $f$ such that $f<1$ near the point $(0,1,0)$ and $f=1$ out of its small neighborhood.