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This paper investigates the exact number of limit cycles given by theaveraging theory of first order for the piecewise smooth integrablenon-Hamiltonian system \begin{eqnarray*} (\dot{x},\ \dot{y})=\begin{cases}(-y(x+a)^2+\varepsilon f^+(x,y),\ x(x+a)^2+\varepsilon g^+(x,y)),\ \ x\geq0,\\(-y(x+b)^2+\varepsilon f^-(x,y),\ x(x+b)^2+\varepsilon g^-(x,y)),\ ~ \, x<0,\\\end{cases}\end{eqnarray*} where $ab\neq 0$, $0<|\varepsilon|\ll 1$, and$f^\pm(x,y)$ and $g^\pm(x,y)$ are polynomials of degree $n$. It is proved thatthe exact number of limit cycles emerging from the period annulus surroundingthe origin is linear depending on $n$ and it is at least twice the associatedestimation of smooth systems.

Limit cycles appearing from perturbations of cubic piecewise smooth center with double invariant real straight lines