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Carath\'eodory's well-known conjecture states that every sufficiently smooth,closed convex surface in three dimensional Euclidean space admits at least twoumbilic points. It has been established that the conjecture is true for allrotationally symmetric surfaces; in this paper, we investigate the umbilicpoints of two families of surfaces without rotational symmetry, and computetheir indices. In particular, we find that the family of surfaces of the form$ax^{2k}+by^{2k}+cz^{2k}=1$ with $a,b,c>0$, $k\in\mathbb{Z}_{>1}$ admit 14umbilic points: six of one known form and eight of another. For many testedvalues of $a,b,c,k$, such umbilic points have indices $-1/2$ and $1$,respectively. We also explore the dependence of the umbilic points on theparameter $\epsilon$ of the surface $ax^2+\epsilon x^4+ay^2+\epsilony^4+bz^2=1$. In particular, for both $a<b$ and $a>b,$ there exist exactly twoumbilic points with index 1 for $\epsilon$ smaller than certain criticalvalues. For larger $\epsilon,$ surfaces with $a>b$ admit exactly ten umbilicpoints; for many tested values of $a,b,\epsilon,$ these points have indices 1/2and -1. For larger $\epsilon,$ surfaces with $a<b$ admit eighteen umbilicpoints; for many tested values of $a,b,\epsilon,$ these points have indices-1/2 and 1.

A Study of the Carath\'eodory Conjecture through Non-Rotationally Symmetric Surfaces