1.1.59 अदर्शनं लोपः

The अनुव्ऋत्ति of this rule is न वेति विभाषा (which makes it appear somewhat out of place). The meaning is simple - a character whose लोप has occurred can disappear.

1.1.58 द्विवचने अचि

The अजादेश caused by द्विवचन is done at the place of स्थानी. The द्वित्व itself can be due to एकाचे द्वे प्रथमस्य. In पपतु:, पा पा अतुस् - results in पा प् अतुस् (अजादेश at स्थानी) and becomes पपतु:.

Note that interpretations of the rule differ in महाभाष्य and वैयाकरणसिद्धान्तकौमुदी. The latter assumes that अजादेश should not occur before द्वित्व is performed. The result (at least in these examples) is not affected because of this difference.

1.1.57 न पदान्तद्विर्वचनवरे यलोपस्वरसवर्णानुस्वारदीर्घजश्चर्विधिषु

The exception rule states that पर्-caused अजादेश is not "applied"in cases when following occur:

1. पदान्त
 अस् तस् - स् तस् - स्त:
If अलोप was स्थानी-based आव् वृद्धि would have occurred. But this being end of पद the स्थानिवत् भाव does not apply.

2. द्विवचन
In the following case, द्वित्व is allowed because यणादेश is not  स्थानिवत् - otherwise अनचि चwould not be applicable.

दध् य त्र - दध् ध् यत्र ( अनचि च )

3. वरे
 या या यङ् - या या य
When above combines with वरच_ then अल्विधि bars स्थानिवत् लोप of अ. The लोप occurs because of वरच_ (due to अच: परस्मिन् पूर्वविधौ) -
या या य + वरच_ - या या य् वर - यायावर:

4. यलोप
 In the following case, if अलोप occurred at स्थानी then कण्डूय्ति would have resulted.

कण्डू यक् - कण्डू य क्तिच् - कण्डू य ति 

5. स्वर
 In the following case, 6.1.187 would have made अक as उदात्त but the current rule bars that and what's उदात्त is ई.

चिकीर्ष अक - चिकीर्षक

6. सवर्ण
 In the following case, स्थानिवत् would have resulted in barring of परसवर्ण due to अनुस्वार. But since there is no स्थानिवत्  due to current rule, the परसवर्ण follows and ण् develops.

शिन् ड_ ढ_इ - शिन् ड_- शिंढि - शिण्ढि

7. अनुस्वार
 स्थानिवत्लोप of अ could have barred अनुस्वार in the following case but the current rule allows अनुस्वार.
शिन् ष् अन्ति - शिंषन्ति

8. दीर्घ
दीर्घ is allowed in the following case because of current rule.
प्रतिदिवन् टा - प्रतिदिव्न् आ - प्रतिदीव्न् आ

9. जश्
In the following case, अच: परस्मिन् पूर्वविधौ would have barred जश्त्व because of झलां जश्झसि. But the current rule let that happen:
घ् धि - ग् धि

10. चर्
खरि च is allowed because of current rule.
अ घ्स् अन् - अ क्स् अन् - अक्षन्