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In this paper we work toward the Homflypt skein module of the lens spaces$L(p,1)$, $\mathcal{S}(L(p,1))$, using braids. In particular, we establish theconnection between $\mathcal{S}({\rm ST})$, the Homflypt skein module of thesolid torus ST, and $\mathcal{S}(L(p,1))$ and arrive at an infinite system,whose solution corresponds to the computation of $\mathcal{S}(L(p,1))$. Westart from the Lambropoulou invariant $X$ for knots and links in ST, theuniversal analogue of the Homflypt polynomial in ST, and a new basis,$\Lambda$, of $\mathcal{S}({\rm ST})$ presented in \cite{DL1}. We show that$\mathcal{S}(L(p,1))$ is obtained from $\mathcal{S}({\rm ST})$ by consideringrelations coming from the performance of braid band moves (bbm) on elements inthe basis $\Lambda$, where the braid band moves are performed on any movingstrand of each element in $\Lambda$. We do that by proving that the system ofequations obtained from diagrams in ST by performing bbm on any moving strandis equivalent to the system obtained if we only consider elements in the basicset $\Lambda$.  The importance of our approach is that it can shed light to the problem ofcomputing skein modules of arbitrary c.c.o. $3$-manifolds, since any$3$-manifold can be obtained by surgery on $S^3$ along unknotted closed curves.The main difficulty of the problem lies in selecting from the infinitum of bandmoves some basic ones and solving the infinite system of equations.

Topological steps toward the Homflypt skein module of the lens spaces $L(p,1)$ via braids