{ "abstract": "Motivated by Zygmund's differentiation conjecture and Stein's conjecture for Lipschitz vector fields, we study bilinear singular integral and maximal operators along variable curves given by $$ \\begin{aligned} BH_a^α(f,g)(x) &= \\operatorname{p.v.}\\int_{\\mathbb R} f(x-t) g\\left(x + a(x)t^α\\right)\\,\\frac{dt}{t},\\\\ BM_a^α(f,g)(x) &= \\sup_{r>0}\\frac{1}{2r}\\int_{-r}^{r} \\left|f(x-t)g\\left(x+a(x)t^α\\right)\\right|\\,dt,\\\\ BSH_c^α(f,g)(x) &= \\operatorname{p.v.} \\int_{\\mathbb R} f(x-t) g(x + t^α)\\, \\frac{dt}{t-c(x)}, \\end{aligned} $$ where $α> 0$ and $a,c:\\mathbb R\\to\\mathbb R$ are measurable. For every $α>0$ and $01}L^s(\\mathbb R)$, one of which is compactly supported, whose outputs under $BH_a^α$ and $BM_a^α$ do not belong to $L^1(\\mathbb R)$. For any measurable $a:\\mathbb R\\to\\mathbb R$, $BM_a^α$ maps $L^p(\\mathbb R) \\times L^q(\\mathbb R)$ boundedly into $L^r(\\mathbb R)$ whenever $1 1$, and maps $L^p(\\mathbb R) \\times L^q(\\mathbb R)$ into weak $L^1(\\mathbb R)$ when $1/p+1/q=1$, with constants independent of $a$. Finally, we investigate the shifted operator $BSH_c^α$, including dyadic-integer-valued $c$, and prove that there is a measurable choice of $c$ such that the operator admits no bounded $L^p(\\mathbb R) \\times L^q(\\mathbb R) \\to L^{r,\\infty}(\\mathbb R)$ estimates in the range $1 \\leq p,q \\leq \\infty$ and $\\frac{1}{2}