I've found two different ways how to do this. The first way is to use XBL to define new component. It can be quite complicated for those developers who has no previous experience with it. Fortunately, there is another way which is more simple.
<textbox id="TB-Search"
minwidth="200" width="200"
onfocus="tb.searchBoxFocus(this);"
onblur="tb.searchBoxBlur(this);"
onkeypress="tb.keyHandler(event);"
>
<image id="TB-SearchImage"/>
</textbox>
CSS definition for search image:
#TB-SearchImage {
list-style-image: url("http://yoursite.com/favicon.ico");
padding-right: 2px;
padding-left: 0px;
margin: 0px;
border: 0px;
}
That's it!