On the similarity of linear operators in $L_p$ to integral operators of the first and second kind

We construct an example of a compact operator of the third kind in $L_p$ ($p\ne2$) not similar to any integral operator of the first or second kind. This example shows that not every linear equation of the third kind in $L_p$ ($p\ne2$) can be reduced by an invertible continuous linear change to an equivalent integral equation of the first or second kind. The example also proves the impossibility of a characterization of integral and Carleman integral operators in $L_p$ ($p\ne2$) in terms of the spectrum and its components.