Unity of invention is defined in Article 82:
The European patent application shall relate to one invention only or
to a group of inventions so linked as to form a single general
inventive concept.
Regarding the independent claims Rule 43 (2) apply:
Without prejudice to Article 82, a European patent application may
contain more than one independent claim in the same category (product,
process, apparatus or use) only if the subject-matter of the
application involves one of the following:
(a) a plurality of interrelated products,
(b) different uses of a product or apparatus,
(c) alternative solutions to a particular problem, where it is
inappropriate to cover these alternatives by a single claim.
Rule 64(1) defines what happens if the application lacks unity:
If the European Patent Office considers that the European patent
application does not comply with the requirement of unity of
invention, it shall draw up a partial search report on those parts of
the application which relate to the invention, or the group of
inventions within the meaning of Article 82, first mentioned in the
claims. It shall inform the applicant that, for the European search
report to cover the other inventions, a further search fee must be
paid, in respect of each invention involved, within a period of two
months. The European search report shall be drawn up for the parts of
the application relating to inventions in respect of which search fees
have been paid.
Rule 137 (1) defines, that:
Before receiving the European search report, the applicant may not
amend the description, claims or drawings of a European patent
application unless otherwise provided.
If I understood the question right, the following claims we filed:
- independent claim - invention 1 - not novel
- depending on claim 1 - invention 1 - novel
- independent claim - invention 2
In this case, the EPO according to Rule 64 sends you a partial search report regarding invention 1 with an invitation for paying an additional search fee for the unsearched invention 2.
2 months after the invitation, a European search report will be drawn up for the inventions for which search fees have been paid.
Before receiving a search report, the applicant may not amend, see Rule 137(1).
In other words, unity is checked before a search report is drawn up. Amendments are only possible in response to Search/Examination reports. (see rule 137(1) and (3))