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This paper is concerned with asymptotic analysis of strongly decaying solutions of the third-order singular differential equation $x^{\text{'}\text{'}\text{'}}+q\left(t\right)x^{-\gamma }=0$, by means of regularly varying functions, where $\gamma$ is a positive constant and $q$ is a positive continuous function on $[a,\infty )$. It is shown that if $q$ is a regularly varying function, then it is possible to establish necessary and sufficient conditions for the existence of slowly varying solutions and regularly varying solutions of (A) which decrease to $0$ as $t\to \infty$ and to acquire...