the order of the integration in (8.15) we see that $||B^{\star}f_{n}||{\to}0,$ i.e. the operator $B^{\star}$ is compact. Hence the operator $B$ is compact too. The proposition is proved.\
Using formulas (8.5) and (8.11) we obtain the assertion.\
**Proposition 8.3** *Let the conditions of Theorem $8.1$ be fulfilled. If the functions $N_{1}(x)$ and $N_{2}(x)$ can have a discontinuity only when $x={\pm}c$ then the function $\Phi(x,y,c)$ can have a discontinuity only when* $x=y.$\
**Corollary 8.1.** *Let the conditions of Proposition $8.3$ be fulfilled. Then the eigenvectors of the corresponding operator $B$ are continuous.*\
We use the following assertion (see\[22\],p.73).\
**Proposition 8.4.** *Let the following conditions be fulfilled.\
1).The kernel $k(x)$ of the operator $S$ has the form* $$k(x)=\mathrm{log}\frac{A}{2|x|}+h(x),\quad -2c{\leq}x{\leq}2c,$$ *where* $ A>0,\quad A{\ne}c.$\
2). $$\int_{-2c}^{2c}|h^{\prime}(u)|^{q}(2c-|u|)du<\infty,\quad
q>2.$$ 3. *The equation* $$Sf=\int_{-c}^{c}[\mathrm{log}\frac{A}{2|x-t|}+h(x-t)]f(t)dt=0$$ *has only the trivial solution in $L^{p}(-c,c)$, where $1/p+1/q=1.$\
Then the equation* $$Sf=g,\quad g^{\prime}(x){\in}L^{p}(-c,c)$$ *has one and only one solution in* $L^{p}(-c,c).$\
**Corollary 8.2.** *Let the conditions of Proposition $8.4$ be valid. Then there exist the functions $N_{1}(x)$ and $N_{2}(x)$ which satisfy the equations* $$SN_{k}=x^{k-1},\quad N_{k}(x){\in}L^{p}(-c,c),\quad
k=1,2.$$ *The functions $N_{1}(x)$ and $N_{2}(x)$ can have a discontinuity only when* $x={\pm}c.$\
**Remark 8.1.** If conditions (7.13) and (7.14) are fulfilled then according to (7.15) 